I watched the film "Tolkien" on Film 4 today, having recorded it recently. I think it might also be on Amazon Prime, for those that have it. I can highly recommend it.
Birmingham obviously features quite often, particularly Barrows (obviously filmed somewhere similar elsewhere) where Tolkien and his friends, who were then at King Edward’s School, met regularly as the "Tea Club, Barrovian Society", named after their Barrows Stores meeting place.