Is it illegal not to carry your driving license when driving?

USA: Yes. You are required to have it on you whenever you drive and show it upon request of law enforcement.

UK: No. It used to be that the police could give you a form commonly known as a "producer", commanding you to produce your driving license (and/or insurance and MOT (car safety test) certificate) within 7 days at a police station of your choice. But I've heard police say they don't give producers any more: all the details can be checked automatically.