My babies nearly 7 weeks and I don’t want to leave him at all with anyone until he is older.
Completely normal. Look up the 4th trimester. Your baby should still be in you. It’s healthy and normal to want to be with him all the time.
My partner thinks I have separation anxiety
That makes no sense at all. Your son and you need each other. It’s nature. Does he often try to undermine or criticise your normal emotions and behaviour?
MIL goes on at him about having the baby, taking him out for a walk etc and then rolls her eyes when he tells her that I won’t leave him with anyone else without me being there
It’s your baby. Not MIL’s baby. She had her turn. No one who rolled their eyes at me would be someone I trusted to care for my newborn. She’s a dick. So is your partner if he puts up with it.
also think he should be left to cry when they want to put him down to sleep etc which I personally don’t agree with when he’s so young
No one who thinks that’s okay would have care of my baby. You can’t trust them because you know they are idiots.
There’s no age when you have to leave your baby to cry. It’s upsetting to hear them cry for a reason. You’re wired to sooth and comfort your child. Normal.
They don’t care about upsetting you (or your baby, given half a chance) so time to stop caring about upsetting them.
A very clear “I won’t be leaving him with anyone until I’m ready. I won’t continue to discuss it because you’re being rude and disrespectful. The more you push me on this the longer it’ll be before I leave him anyone”. “Stop asking, I’ve said no”. “He’s my baby, I’ll decide when I want to be apart from him”. And get up and walk away or ask them to leave, put them phone down etc.
Now is the time to draw your boundaries. If they continue to try and bully or undermine you then stop seeing them.