I think the houses may have been completed, or certainly started before WW1.
I am not very well up on military matters, and perhaps those that are could comment on the following.
In c1905 and before the area of the drill hall would have been in Worcestershire (Kings Norton district) and so presumably the drill hall would have been for the Worcestershire Volunteers. Up till about 1890 the drill hall of the Worcestershire (1st) Artillery volunteers (headquarters of 5,6,11 & 12 batteries) was shown on maps and in Kellys as in Cox St West. Perhaps the later drill hall replaced this one. The dates would fit. One of the reasons that it disappeared may have been because of the incorporation of the area into Birmingham. Do any of these batteries correspond to people, mentioned on the plaque