I hate to say it but cyclists palsy (ulnar nerve damage) is a common problem. You will need to take precautions to avoid it for example-

Padded gloves (make sure the padding is right for you, some like lots, and some find minimal works for them).
Bar tape with padding
Carbon fork and/or handlebars to reduce road buzz
Moving hands often and using different hand positions
Making sure your gloves/watch is not putting pressure on your wrists
Trying to avoid bumpy roads (yeah right!) and making sure you limit the time on the bike to what you can stand without causing your arm nerves to play up.
Keeping the weight off your hands- strenghtening your core, raising your handlebars etc.

Good luck. I know nerves can regenerate, but very slowly.