I haven't actually ran a game yet, but I think this is a really nice approach to take with RoL, especially with players who haven't read the books (and even with those who have to be honest.)
I think you could work the required downtime into the game fairly easily, having their main duties take up time - nurses or other emergency service workers spending weeks on long shifts or overtime consuming big chunks of time, only leaving enough for sleeping and some light research before the next game session takes place would be a believable example. The boxer having to prepare for a major fight, going off to a training camp could be another.
I'm sure in one of the books Peter talks about his training routine, mentioning a couple of hours practice a day maximum for learning magic forma, the rest being reading, translating stuff from Latin to English etc.. So I think there are definitely ways that you can work their training into the narrative and have time progress for all the characters in some kind of montage effect, with some working while others learn spells and other required skills.