very easy to over winter begonia tubers, take them in before the frost, if you have a greenhouse they can continue flowering inside, I let mine die back naturaly this way the good ness goes back into the tuber no watering.
Once all top growth has died off remove and clean tuber of old peat, store in fresh dry peat ( this way if vine wevel is present you get rid of them) frost free untill spring bring out and start in damp peat trays and off you go again.