She just wants to take over with your baby.
Push back, politely and very very firmly, and keep pushing - until she gets the message. Long term, your relationship will be better for it (for 'better', read 'possible at all'!)
She wants to be the boss of her son's family as well as her own - to stay in pole position as matriarch and 'mum figure', as she always has been where he is concerned.
So - that's fine to sort, as long as you aren't a pushover. The potential bigger problem is your fiance. It's understandable that a situation like this is upsetting for him - but he needs to understand VERY clearly who exactly is causing it and how it will most easily get sorted, because what his mum will be pouring into his ear will be the very opposite.
'DF, this is really about your mum wanting to feel she is in control. It's about you growing up and getting your own family and her feeling sidelined by that change. It's about her wanting to still feel she's the 'mum figure' and trying to maintain that by taking over with our baby.
That can't and won't be happening, not if we want to continue to be happy together - I'm not going to let someone else, no matter who they are, dictate or even be INVOLVED in decisions that are ours, and I certainly won't be stepping aside and letting your mum take over with our baby. But, I also want a good relationship with her and I definitely want her to feel she's and important part of all our lives. The best way for that to happen is for us to present a united front - it will be difficult at first, but the more mixed messages she gets, the more she will guilt trip and try and push things, and the result will be that I fall out with her. The result WON'T be that she gets what she wants. I don't want to fall out with her and I don't want us to fight about it. The best - the only way to handle this is for us both to be very firm and clear that we'll be making the decisions, we'll be doing the parenting, and if she wants to be involved, then she needs to respect us both AS THE PARENTS.
'I suggest you contact her and let her know we love and value her but we will absolutely not be discussing childcare arrangements or any other aspect of how we arrange our lives with her - they are decisions for us alone. She needs to respect that just as we respect her decisions about her own life. Also, we will NOT be having anyone to stay for extended periods in our home.'