(Ah, wondered when you would notice that one, Wilson!)
That's one of the little problems, Phil. After the flurry of forming the different battalions initially, there was a lot of reorganising and redistribution. The ten or so first Birmingham battalions extended to over thirty, eventually, and in some (?all) cases got new numbers. The 5th Birmingham Battn. became the 25th Warwickshire (Birmingham) Battalion, for example. The 5th Warwickshire was something else.
It's almost certain that the 3rd Birmingham became the 23rd Warwickshire (Birmingham) - the C.O. of each was the same chap who is taking the salute in the pic.
BUT... the 23rd Warwickshire (Birmingham) was based in the Aston area. This tends to argue against the Bristol Road South identification (even though from what I remember of occasional tram journeys in the early/mid 1940s, clattering down those long straight stretches at an exciting top speed, it certainly looks a good call). And it does make one wonder about Derek's Tyburn Road suggestion.
Wish we could read the shop names!
Chris