Firstly, to his mother, you and she are not 'equal'. You are a current girlfriend who may be around in a year, may not. She is the mother of her grandchild, a member of the family. Right now and pretty much unless you and her son also had a child together, the ex is, and is going to stay, a lot more important to his mum than you are.
And, 'clinging on'? No, she's building a new relationship with her DD's grandmother/her MIL, now that the original link is severed/altered. She has to do this on some level; if she wants to have a harmonious wider family life for her daughter; if she's always been closer to her MIL than her mum etc then it's going to matter to her on a personal level too. The joint link was your BF, now it's the child.
You MUST see this differently or you'll end up miserable, and if you try and influence things, it will be very hard to come back from - you'll be resented from all angles and (quite reasonably) be seen as poking into relationships which aren't your business.
Super sensible points here.
The fact is even if they're being bitchy and slagging her off, your partner and his mum are on board with the relationships as they are facilitating it.
A year is not long at all, you are a relatively new girlfriend and as PP said she is the mother of his child and will definitely be in his life forever.
Phrases like 'weekend mum' are so dripping in judgement it's laughable. You know full well you meant she's not a very good mum, which is such a shitty thing to say about someone when you don't really know the background fully.
You only have your boyfriends side of the story, as does his mum to an extent.
You sound too immature to deal with what is a complicated dynamic that requires people to put this child first. It's in their interest to have close relationships with family especially if they only have them on one side.
It's petty to bring up she cheated, if she did. That's utterly irrelevant to them coparenting now or your boyfriends mum being the little ones grandma.
Children come first. If you can't deal with it you can't date someone who's married. A year is early to live together when there are complications like this.
His mum might not be that bothered about getting to know you yet because she doesn't know if you'll be a long term girlfriend yet and it sounds like she's really busy helping your boyfriend and his ex out to be honest.