I hope people haven't forgotten the excellent publication from Christopher Phillips: Birmingham Votes 1911-2000. It was issued by LGC Communications in 2000 under the auspices of the LGC Elections Centre in Plymouth.
At over 260 pages, it contains the results of every election, including by-elections, during the period. There are short commentaries about each borough election, statistics, maps etc. The results are presented in two ways: Firstly, for each (usually annual) election, listing the ward results alphabetically by ward name. Later in the book each ward is examined individually by providing a chronological listing of results over the complete time-frame, but in a more concise style so as not to make unnecessary repetitions from the first section.
There is also Chapter 5 in Sam Davies' and Bob Morley's first volume of County Borough Elections in England and Wales, 1919-1938 to remember. Obviously the period studied by them is more narrow and they do not give the results of by-elections or when the poll was postponed. The book from Philllips is therefore much to be preferred as the source of results for Birmingham elections.