My situation - 20 odd years ago - she moved out, joint mortgage although i was paying it all.......that made no difference legally, i couldn't ask her to contribute half. i had to pay her off as she could've forced a sale but luckily i got away with a one off payment which she took. Still stuck in my throat to pay her (she was unfaithful)
I changed the locks but apparently, i shouldnt have - took a chance but got away with it. I dont know if that's still a thin ice thing to do but u was preventing her access to her possessions etc. Got a bollocking off the solicitor for it.
Will say, no kids were involved so i cant help you on that bit. Dont know if CSA are still a toothless tiger of an agency.
If you stand your ground and stay - which i believe is your right - then prepare for things to get shitty. i really played my face up about anything just to be awkward. In the end i thought life is too short and i said sod it and co operated.
Dont know of overcrowding is a problem at either parents house but my guessing is she wants her nest back to bring the babby up so wants you out as it would suit her more.