air travel – Can I bring a single shot of live ammo onto the plane from US to UK as a souvenir?

It is not permitted to bring ammunition into the UK without a permit. According to this brochure from the UK Border Force (p. 14–15):

There are certain goods you are not allowed to bring into the UK – this is to protect society, animal and public health and the environment.

The following are also banned but in certain cases may be brought into the UK if you have obtained the relevant licence or permit:

Firearms, explosives and ammunition

It is unlikely that it will be worth your while to obtain the proper permits simply to bring one round of ammunition into the UK.

A spent cartridge might be a better souvenir, though even then you should check with your airline to be sure there aren’t any rules against it or special procedures that you’ll have to follow.