I've never heard of a climbing begonia. Trailing, but not climbing. Have you tried an internet search?

You may be thinking of Cissus discolor, a tendril climber, commonly known as Rex begonia vine. It's tender and is really a houseplant here in the UK - not sure if you'll even be able to find one in the UK - its common in the States though, info and images from an American site below

http://awaytogarden.com/the-rex-begonia-vine-cissus-discolor/