Before I start this, I'd like to point out that I took significantly less time to answer this question than it took yayfulness to answer mine. I'd also like to apologize for errors in transcription. I was tired, it was taking longer than I expected, and I really only wanted to include pertinent information. This is not an official transcription. Additionally, one of the journal writers didn't believe in complete sentences or really punctuation or capitalization. One final apology: I didn't look at Margetts' trip back across the plains, which I'm sure was recorded. Once again, it had been an hour and a half of transcribing at that point. But this should be enough information.
Alright, so your first parameter of this question is that the missionaries had to be sent out between 1847 (the year the Saints arrived in the Salt Lake Valley) and 1869 (when the Transcontinental Railroad was completed). But even while the railroad was being constructed, it was sort of in use, making it dependent on the year as to how far they would've had to travel.
Since our only source for this information really is missionary journals, that's what I've turned to. Unfortunately, there are very few missionary journals from this time period that I have access to. It's entirely possible that the Church History Library has more, but since I'm in Provo and it's in Salt Lake City, I have to depend on Special Collections instead.
Luckily, there were two resources for this. One was a microfilm, and the other was available as a microfilm (as well as available as his original journal).
Our first resource is from John Lyman Smith. He was called on a mission to England in 1855 and on a mission to Europe in 1860, both dates falling within the specified timeline.
On page 22 of his journal, he begins to record his mission to England:
April 6th, 1855 at a general conference held in Great Salt Lake City, I was appointed to take a mission to Europe. April 28th I was set apart by Elders Orson Hyde, Orson Pratt, George A. Smith and Wilford Woodruff at my house in G.S.L. City, being ill with the rheumatism and unable to turn myself in bed. On May 7th President Brigham Young called and blessed me and said I would begin to amend from that very hour. At 2 pm I was lifted into the wagon and laid on a bed being unable to sit up. Thus I started for Europe. While going down East Canyon Creek, the wagon was overturned. I was still on a bed and unable to help myself. At Fort Bridger the horses ran away and the wagon was only saved from being dashed in pieces by the horses breaking away from it. June 20, we met Seth M. Blair’s company from Texas…We laid hands on several of them and pursued our journey to Atchison, or Mormon Grove, on the 21st. On the 22nd in company with Elder David A. Curtis, I started for St. Louis, leaving the other brethren to sell our part of the horses and wagons of our outfit. On the 26th I arrived at St. Louis in the morning. On June 27 we took steamer for Keekuk, where we arrived on the 28th.
On page 28, he records his travels back from England in 1958:
Arrived in New York March 10, and went to Burlington Iowa, remained a few days, and then went to Florence and waited one month for the arrival of the Elders from the east. On May 2, 1958, I started from Florence (Winter Quarters) for Great Salt Lake City, having been absent three years and forty five days. The last day I walked fifty(?)-five miles.”
On page 34, he records his travels to Europe in 1860:
September 22nd, 1860. Salt Lake City. At the Historian Office, I received from under the hands of seven of the Twelve Apostles, a blessing and setting apart for a mission to Europe...My cousin Jesse W. Smith, William W. Cluff, John H.? P. Johnson, and myself arranged a light wagon, one horse and harness each, provisions, bedding etc., reading for starting; and on the evening of the 25th we attended a party at the social hall under the auspices of the presidency, especially in behalf of the missionaries going east...Wednesday 26th we bade adieu, received the parting blessings of all and at two pm started for Europe. Before reaching the Weber river the company chose Claudius V. Spencer captain of travel under the direction of the Apostles three of whom were with us. Some twelve or fourteen wagons were in company. October 3rd 1860. -- We passed Fort Bridger and on the 8th crossed over the South pass. Upon the 18th we passed Fort Laramie at 2 pm. The road is lined with teams en route for Denver and the Pike's peak mines. We arrived at Florence (our old "Winter Quarters") on the 6th of November. On the 13th the Elders had all gone except Elder John T. Gerber who remained with me as his mission is to be with me in Switzerland. Brother William D. Johnson took me in his spring wagon to Omaha and we took passage by steamboat "Chippoway" for St. Joseph Mo. Where we arrived on the 18th; took railroad train for Palmyra, 192 miles, crossing river Mississippi...
On page 76, he records his journey back across the plains in 1864:
Wednesday, 27th - We arrived at Quincy at twelve noon. At six pm we ferried the Mississippi river. Here we received a dispatch that Salt river bridge and Shelbine(?) station on the Hannibal and St. Joseph railroad had been burned by guerillas and we had to camp in the woods near the station. Thursday, 28th - Three trains were got in readiness and we reached the vicinity of the burned bridge at noon. We left the train and forded salt river and camped in the woods. Our luggage had to be conveyed across the river three fourths of a mile, mostly on men's backs, as only three wagons were obtainable for the heavier packages...Saturday, 20th - We all arrived at St. Joseph Mo. The last train at three pm. The roughest railroad ride I ever experienced. Sunday, 31st - After much delay, trouble and bother we left ...
Now, before I summarize all of this information, I'm going to provide for you one more source. Bear with me.
Phillip Margetts was called on a mission to England in 1857. On page 4 to page 30, he describes his journey across the plains. I've transcribed a lot of it here. Once again, I wasn't entirely sure on some words, and there are some dates where I just didn't know what to transcribe, so I didn't. (I'm sorry. It really was getting difficult to concentrate at that point.)
On the 23rd of April 1857 I started on Company with 76 missionaries from Great Salt Lake City to nearly all parts of the world, myself bound for England. This morning we started with handcarts instead of teams, hundreds of saints following us out of the City, when on the bench(?) we had the misfortune to bend our axel tree(?), sent it back to the City and in two hours we were ready to depart(?) again and traveled 2 1/2 miles of Emigration Canyon and camped for the night...
April 24th. Arose this morning and after breakfast we organized ourselves into a company...and proceeded up the Canyon and acsended the little mountain first ? and camped to noon at the willow things, after which we proceeded to the foot of the Big mountain and camped for the night where it was very cold being considerable snow.
April 25. After a very hard hill(?) we succeeded in getting to the top of the big mountain where we found about 12 feed of snow...and traveled down the mountain and ate dinner on the first crossing of Canyon Creek. We crossed the creek several times upset our cart and camped at the sixth crossing of the creek.
Sunday, April 26th arose this morning all well and ate a good breakfast than traveled along and crossed the creek ascended (?) hill and then went at a good speed and arrived at Weber river and by the end aid of the teams we crossed first late and camped on the east bank.
April 27. Arose at 4 o'clock...packed up and traveled 5 miles before breakfast. After...we went 9 miles and rested till the wagon arrived with the provisions then traveled 5 miles farther and camped. traveled today 19 miles -- all well.
April 28th. Camp called at 4 o'clock traveled 5 miles and ate breakfast, then went over to Yellow creek for dinner after resting a short time we started again and arrived at bear river about 6 o'clock where it commenced to snow and continued all night, ell about 6 inches.
April 29th we arose this morning all well, snow almost 4 inches deep, after breakfast it commenced hailing, we traveled 6 miles and stopped for dinner as the (??) on the (?) after dinner it cleared up, we then assembled the hill and traveled 8(?) miles then camped for the night at Soda Springs.
April 30th. The ground was covered with snow this morning about 4 inches deep, after breakfast traveled to muddy and managed to get and cross without wetting my feet we made a campfire and after a good warm by our fire, we proceeded to Fort Bridger after ascending the Rocky patch, which was not accomplished without a long and a strong pull, we arrived at the fort about 7 o'clock, the day was very cold.
May 1st...ate our dinner and started from Fort Bridger at 2 o'clock traveled over a beautiful road and arrived at Smith's Fork at 6 o'clock traveled today 12 miles
May 2nd we traveled 5 miles and crossed Black's Fork 2nd time before eating breakfast this morning the water was tremendous cold ice about 1/4 inch thick the banks was likewise froze which almost froze our feet after walking 12 miles we ate dinner. we then crossed "Thames" fork were 6 miles and then camped.
May 3rd after breakfast we traveled 17 miles and crossed green river which was about two feet six inches deep...we all got over safe and camped for the night on the east bank, held a meeting in the evening...
May 4th this morning we crossed the green river cut off, ate dinner on big sapay(?) after dinner we traveled 17 miles to the next crossing of Big Sapay(?) and camped. traveled today 29 miles
May 5 traveled this morning eight miles before breakfast camped on little Sapay(?) traveled fourteen miles to Dry Sandy (?)
May 6th After breakfast we walked eleven miles and ate dinner on the "South Pass" or "Perifia Springs"(?) after which we crossed the Pass altitude (????) above the level of the sea traveled onto the first crossing of sweet water took some wagons and got a fresh supply of provisions...traveled today 22 miles
May 7th camp called at four o'clock traveled five miles on the (?) cut off before breakfast then went on twelve miles and stopped for dinner on the rocky ridge wind blowing...after which hurried on four miles and camped for the night traveled today 21 miles
May 8th this morning it was very cold traveled 6 miles...traveled today 30 miles
May 9th traveled today 28 miles
May 10th after hauling our carts 4 miles we ate breakfast on deep creek, 7 miles from "Devils Gate" after eating we hitched up and in two hours we all arrived safe at the above named place found all at the fort well...
May 11th this morning still as the devils gate after eating I repaired our cart and got everything ready for starting we left the place at 12 o'clock...then traveled 15 miles
May 12th camp called at four o'clock after traveling 4 miles we ate our breakfast on a little creek...then traveled 8 miles to willow creek this morning bro Mcintosh broke the axel of his cart which was (?) divided the food and hauled the empty cart travled ten miles and then camped for the night
May 13th we traveled this morning 5 miles to Alkaili Flats...after which we traveled 12 miles and camped for noon while eating dinner ait commenced to storm we rested 2 hours and proceeded to Platt Bridge arrived there about half past six o'clock and camped on the north side all night
May 14th after working about 4 hours in the blacksmiths shop we started down the south side of platt and arrived at muddy creek 6 miles from platt bridge ate dinner went 10 miles down the river and in consequence of thunder storm we camped for the night
May 15th traveled 6 miles and ate breakfast ...traveled 9 miles...
May 16th traveled today 23 miles
May 17th this morning we traveled five miles and crossed the La Bonte river this day we made about 25 miles
May 18th after break fast we went over to horse shoe creek where we found Porter Rockwell
May 19th traveled 6 miles and ate dinner...Better cotton wood is 3 miles from here
May 20th. After eating breakfast we traveled through the black hills 12 miles and ate dinner on Platt 12 1/2 miles from Fort Laramie after traveling 4 miles we came to a blacksmith we then went 8 miles and arrived safe at the ferry at Fort Laramie 509 miles from the valley
May 21st Got supplies from the fort...then we traveled ten miles
May 22nd Traveled 7 and a half miles to raw hide creek ate breakfast and after we traveled for about 9 miles and camped for dinner...after dinner moved on 5 1/4 miles and camped for the night
May 23rd camp called at 4 o'clock and traveled five miles before eating breakfast, we went ten miles for dinner...traveled today about 29 miles
May 24th traveled today 29 miles
May 25th this day we made 25 3/4 miles
May 26th traveled 28 miles
May 27th made a distance of 28 miles today...
May 28th all tho we made 24 miles we crossed 10 creeks today
May 29th traveled 29 miles today
May 30th we traveled 4 miles and camped for breakfast...
May 31st traveled today 26 3/4 miles
June 1st Bro Richardson Shot 2 buffalo...traveled today 28 miles
June 2nd traveled this day 29 miles
June 3rd traveled today 30 miles
June 4th traveled today 28 miles
June 5th traveled today about 35 miles
June 8th traveled today 31 miles
June 9th traveled today 30 miles
June 10th our journey on the plains is at an end
Alright, this is the part where I give you all of the information that you actually wanted.
It looks like they actually did walk on Sundays. (In a seven day period, there is never a "rest" day. I don't know why.) It also appears that they actually did use wagons, or as in Margetts' case, they used handcarts. This gives them the ability to carry food with them on their journey, and they mention getting provisions from forts. Also, they seem to have hunted along the way, as Margetts mentions two buffalo being shot.
From Margetts' journal, it took them 28 days to get to Fort Laramie, which was 509 miles from the valley, giving them an average walking speed of 18.18 miles per day. This increased to 21.2 miles per day after they left Fort Laramie. It seems that J. L. Smith's group took a lot longer, though they were using wagons more so than handcarts, and he was ill. On his way back, however, it seems J. L. Smith walked almost the entire way. Unfortunately, he didn't provide any details as to how he dealt with food. His second trip out to Europe took about the same time that Margetts' group did.
J. L. Smith's second trip back from Europe involved him traveling with a group of pioneers, so they traveled at the same pace as a normal group. It's entirely possible that this is what happened with Margetts as well.
So. Margetts' journey took altogether 49 days and covered just over 1000 miles, giving them an average walking speed of 22.44 miles per day.
J.L. Smith's first journey took 78 days, giving them an average speed of 14.1 miles per day. His journey back took approximately 51 days, giving him an average walking speed of 21.57 miles per day. His second journey took approximately 23 days, giving him an average walking speed of 43.42 miles per day.
John Lyman Smith. Papers (Typescript and Handwriting). MSS Film 920 #89. L. Tom Perry Special Collections, Harold B. Lee Library, Brigham Young University, Provo, Utah.
I used the typescript of his journal, which was found after the Levi Savage biography on the same microfilm.
Phillip Margetts. Journal (Typescript). MSS Film 920 #56. L. Tom Perry Special Collections, Harold B. Lee Library, Brigham Young University, Provo, Utah.
There's actually a microfilm of his journal available in the Family History department, and his papers are also available in Special Collections.