Well I found this (
http://www.sfmuseum.org/hist1/pxpress.html) about the Pony Express. According to it, Wells Fargo took 20 days to get from St. Joseph, Missouri to Sacramento while the Pony Express could do it in 10 days.
Check up on railroads but I think this was well before Union Pacific and Central Pacific met up (1869?) so however long it took to get from London to somewhere on the East Coast (a couple of weeks for steam ships?) then maybe a day for the train to deliver from New York (probably) to St Joseph. So, maybe 30 days for a letter to get from London to Sacramento?
Unfortunately that has nothing to do with your questions.
Dodge City to New York, a couple of days if the railroad connected the two by then. I have no idea how fast the trains travelled (probably around 60 mph), the exact distance and how long. Ditto for New York to Dodge City I guess. The American No. 1, a 4-4-0, debuted in 1832 so its certainly possible (
http://www.sdrm.org/history/timeline/).
London to Dodge City might be what I suggested earlier. A couple of weeks to cross the ocean, maybe less, then another day or two to get there.
According to this site (
http://www.e-referencedesk.com/resources/almanac/kansas.html), Kansas is 400 miles wide so figure your walking or riding speeds. So quoting from the Pony Express site above:
"A rider would pick it from his predecessor and ride forth sixty miles at top speed to the point where his "relief" awaited him, to pick up the mochilla and start off in turn upon his sixty mile stretch. Six hour were given each of these riders for his sixty mile stint, and in this time he rode six different ponies."
That comes out to around 40 hours to cross Kansas, top speed on horseback, unless I did my math wrong but that seems kinda slow.