Never heard anyone say "mum" unless they were just kinda slurring the word mom. XD I'm from the Midwest. We just say "mom" or "ma" up here, pronounced as "mawm" or even "maam" depending on how far North you go. Never mother because it sounds a little too formal for casual convo. Almost sounds like you're using an uppity tone to call your mom "mother" haha.
I would say the use of "mom" or "mother" could actually say a lot about your character's relationship TO their mother, by that logic.
I know of biddy, but never used it in casual conversation or heard anyone use it casually. Only know it because of the internet, where I picked up all sorts of random words and shit from all over the place. Locally? Nope, never.
I'm super curious to see what other words we could come up with for this haha, feel free to msg if you ever want to ask about words used in the Midwest of the US. =] Good luck with writing!