Probably. I think the creators meant for him to be American, but I prefer to think of him as British, and when you think about it, it makes more sense than him being American.
For one thing, he speaks with what is known as a Mid-Atlantic accent, which is an invented (not natural) accent that was once very popular in theater in the old days: en.wikipedia.org/wiki/Mid-Atla… Bob is also very knowledgeable of Shakespeare, and his mother is a Dame, which is the title of a woman who has been knighted. People who are not native to the UK can be knighted too, but it is very uncommon for people outside of the UK to use the title "Sir" or "Dame" before their name. This implies that Bob's mother, Dame Judith Underdunk, is British, even if Bob is not.