OF 3971, car no. 339, an AEC 'Regent', according to Birmingham Buses at work (David Harvey), was one of thirty similar buses (OF 3970 - OF 3999, car nos. 338 - 367) introduced in 1929. Theey did, it seems have bodywork by Brush. It was the second of the newer type buses with enclosed drivers cab and staircase. Some of these buses lasted until 1950 and a few got the severe wartime bodies fitted later in their lives. 338 and 339 had replacement chassis in 1930 and were re-registered as OG 3638/9. 338 got the wartime body in 1944 and I would guess that 339 also did.
339 has the distinction, it seems, of becoming a driver training vehicle.