This is getting complicated. The hall on Phil's map seems to be called Bloomsbury Mission Hall & sunday School up to about 1964, after which it disappeared in the sensitive replanning of Birmingham.
However, going back to no 307a. In 1890 it is listed as Nechells Hall, with the Manchester society of Oddfellows local branch listed there also. In In 1910 this is listed as Oddfellows Hall (Manchester unity). In 1912, 1913 and 1915 it is Nechells Picture Palace (I didn;t look in those years previously, expecting it to be there a long time). In 1921 nothing is listed there, but in 1932 there is a butcher at 307, but a billiafrd hall at 305 , so I reckon they possibly altered the entrance to the hall down a bit. This was similar till 1950, but in 1955 the billiard hall had disappeared and 305&307 were the east Birmingham Constitutional club (with butcher in front at 307). this continued there till demolition.
So from what has been said by others, I would guess that , although no 290 was officially known as th Bloomsbury Mission Hall, it was locally known as Nechells Hall