The answer is really quite simple and it is very much the patient's responsibility to make this decision.
You should certainly not drive a motor vehicle 24 hours after a general anaesthetic or sedation.
Specifically regarding the hand, for a patient to be able to drive after a hand or wrist injury or surgery they must be able to control their vehicle in the event of emergency. This includes the ability to swerve out of the way of an impending accident such as a dog running into the road.
If you have any concerns please do not hesitate to contact me or indeed your insurance company.