Winter Cage Cleaning?

I have two untamed atiels. I have tried every possible training method, on the internet, bought books etc, and it has never worked. Usually, when I clean the cage in summer, I remove all appendages, clean them, put them back, and rinse down the cage with birds inside so they get a wash too. However, this winter has been exceptionally cold, and I've been refraining from cleaning for a while until the cold snap pes, but it hasn't. I really need to clean the cage soon, but don't know how to do so without getting the birds wet or cold. I can't even transfer them to another cage, as they are that afraid of hands.