So... a few thoughts.
"She was ok then but since having the baby has been rude and distant and keeps my brother and her baby from my parents and the rest of us."
3 months is still a very new baby, still very recent for a mum to be recovering from birth especially surgery. You've no idea what her mental state is. And I'd say this is the reason why .. :
"My mother took 2 weeks off work to stay with her son and help dil for the birth, but she refused and told my brother to tell her that she doesn't want anyone there. "
Why didn't your family respect her wishes? I love my in laws but I couldn't think of anything worse than having them in my home when I was recovering from birth with raging hormones etc. Your family prioritised seeing the new baby over the welfare of the mum, why would she thank you for that?
"We then found out that her own mother was staying at their house."
when you're feeling vulnerable you just want your close family especially your mum. She barely knew you, hadn't even met your family prior to the wedding. Why would she want almost strangers in her house when she's at her most vulnerable instead of her own mum. Especially when they can't respect her wishes.
"My sister was going around in the evenings after the baby was born and she was just so rude ...I'm sure she was a bit irritable after surgery,"
After my section I was exhausted and in pain in the evenings as my pain relief wore off and the movement over the day built up. I HATED evening guests because it meant I had to host them and I couldn't do what I needed to do which was lying in bed with my baby resting. Insisting on going round in the evenings is a massive imposition on a new mum recovering from surgery. You say your family only wanted to help and I'm sure that's true, but sometimes help looks like respecting someone's wishes and listening to them when they tell you they need space.
" but has been 3 months now and still not letting us have access to the baby.
She takes the baby to her mum and dad and sister all the time, but we are made to feel like we can't go around to my brothers house. We still do, but we are clearly not welcome."
3 months would not be long enough for me to move on from someone enforcing themselves on me so they can see my baby at the expense of my welfare. My ds is 11 months now and I'm still cross at my inlaws for turning up multiple times in the first week after I expressly told them we weren't having visitors in the first week. You have impacted on her experience of having a baby and bonding with her child, you've shown her that you don't respect her, her boundaries or her wishes and you are expecting her to just get over it already. She's never going to forget your treatment of her so don't expect her to.
"Her family have lots of grandchildren and mine only have this one so she is being so selfish by withholding him. How can we build a relationship if we aren't allowed to see him."
If someone repeatedly ignored and went against my wishes as mother, I wouldn't trust them with my child.
She is pulling my brother away from his family and being selfish with my nephew.
AIBU?
yes- very unreasonable.
What can we do to get her to see what's she's doing?
You don't, you take a step back and think about how you've come across, how your family's actions have hurt her and made her uncomfortable and then you try and make amends by being more respectful in future.
I understand what you've said in other posts about a bit of a culture clash, but it feels like you're really only interested in seeing this from your perspective without being willing to accept that you've really imposed and over stepped with this woman when she's already been feeling at her most low and vulnerable. If you seriously want to resolve this with her, then I'd go to her and say listen, on reflection I think our attempts to help have maybe crossed a line and made things more difficult for you. That wasn't our intention but we recognise that's what happened and I'm sorry. We do respect and care about you and we want to get to know you better, can we arrange meeting up more regularly as we would love to see more of you all.
You aren't entitled to access to the child if you can't show respect the mother and while naturally you are going off your own traditions, in those moments, it's important to be aware of the traditions and needs of the mother first and foremost because she's the one who's just been through something massive and is still very much raw from that experience. So you follow her lead above your own.