Shouldn't be a problem. You can always use a glove or something and pick the babies up in a bundle, including some of the bedding, place in a bucket with mom, clean cage, and be done with it. Mom mice usually keep their nest clean, so you shouldn't have to worry about that for awhile. I would say you could pick them up individually next week and just clean everything. I never had a problem with gerbils, mice, rats, etc picking up their babies to handle or to clean the cage.