I've found that mice will ALWAYS give birth right before you are scheduled to clean their cages out. I think they're psychic about it. =P
Anyway, I've found that if you leave the nest area and 2 inches on all sides of the nest and then just clean out the rest, and just replace everything else they do pretty good. As long as you leave the familiar scents and whatnot alone, I've not had a problem changing the litter. Sometimes I can reach in and grab the entire nest; babies, material, and all in one handfull and let it sit in a bowl till I'm done, then put it back afterwards.
One of my colonies is so sensitive about their babies that if you even get in the nest and root around searching for the right sized baby, they'll spend hours packing the babies from one spot to another. Usually it's one female who'll insist on the babies being in the food bowl, and the others spend their time putting the babies back in the nest. I still haven't figured that one out.
But I kind of operate under the assumption that they're going to be disturbed for various things throughout the course of their lives...might as well start now.