She ceased to be a British citizen upon her coronation (the monarch is specifically not a citizen), so she wouldn’t have been entitled to one.

If a monarch hypothetically fled or was deposed, they would legally be stateless.

I think she was one of the few people to be exempt from laws as well- abroad she had diplomatic immunity and at home she had sovereign immunity, since all punishments are passed by the monarch.

For Charles III it’s the exact same situation