As a DIL, i'd try to build some bridges with her. Giving advice to a new mum is risky. I was was given so much 'advice' i wanted to scream ' let me do it MY way!'
Whilst they knew you was coming to the hospital, you didnt wait for an invite, you just turned up. That is not fair on your DIL at all. Do you remember what is like to give birth? And all the other horrible stuff that follows? i would of been horrified.
And no. You cant just 'ring and pop round' when you feel like it, because tbh, that also isn't fair on your DIL and SIL. I have a baby. My house aswell as myself are a mess. I don't really have time to eat. Let alone tidy up, iron, hoover, etc. I wouldn't want anyone just poppin in.
How far does your family live? Is it a few hours? Traveling with a baby isn't fun. Do they offer to visit your SIL and DIL?
You said you've already had words and not got off on the right foot. You said something and then your SIL told his wife what you said about her? You seem like you want your DGD all to yourself, even though your DIL isn't comfortable with that.
You said they moaned at you for buying too much for the baby. Did you buy pretty much everything? Like, there was nothing left for then to buy?
She has already told you she feels like 'the oven'
My MIL made me feel like that. She spoke about me behind my back and would go in a strop when things wasn't about her. We don't see her anymore.
Please don't be offended by what i have said. Please have a good think and put yourself in DILs shoes.