does it sound natural?

The habit is to use was or am depending on when the event happened rather than when you are telling the story. The event was in the past so it gets the usage of the past. It is much the same as if you were telling us how hungry you told them you were. "I told them I was hungry." Even though you may have said to them "I am hungry."

If you're specifically quoting what you told the police, you should say "I am British." If you're just talking about what information you passed onto the police at that moment, say "I was British."