My understanding...
up and downs....1700- 1860s
muley ......1850-1860s
Circular .....1860- now
Band saw .....? replacing circular

However seeing as there are little private mills of all sorts don't rely heavily on the saw marks to date a building. Our mill has hewn posts, some vertical sawn main beams, some circular sawn timbers and roof deck (all built in 1950s)

I think one of the main differences with the muley was that the saw was outside the carriage, so the boards could be cut of and the carriage reset to cut the next one off, ....our up and down has to be reset for each cut manually.
Circular was MUCH faster...thus replaced up and down style
Band saws have a smaller kerf (less waste) and can be run faster thus replacing circular.