I believe that the fruiting bodies (mushrooms) that already have sprouted will probably be effected by the weather and get a bit mushy if left in snow/wet environment if not picked but the mycelium underneath the ground will be fine. I would expect that you would get future flushes.
I run into the problem of identifying a patch of small Winecaps that I want to pick, forgetting them for what feels like a day, and realizing they are HUGE after.