Hello Sue R,
A check on FreeBMD for ARNOLD births in Tamworth, mother's name ARNOLD (mother's surname was indexed starting in 1911) from 1911 to 1940, brings up quite a few.
Depending on how much you are able to spend on this quest, it might be worth getting some of the post-1920 birth certs (specifying that the mother must be Annie or variant thereof)? If the mother in any of these later births is an Annie - especially if she has the same occupation and address as your grandmother - then that would be a useful clue ...
Unfortunately it's always possible, of course, that if Annie survived she might have married or moved away from Tamworth after your mother's birth ...
However, as a first step, have you checked through the full death indexes for your grandmother Annie's possible death in childbirth (nothing having been found on FreeBMD)? If nothing can be found in the full indexes in Tamworth in the same year (should be the same quarter though perhaps extend the search to the following quarter) as your mother's birth, that surely would put paid to the idea of Gma Annie dying in childbirth ...
Very best of luck in your quest,
JAP