Can anyone sympathise or give advice?
Yes.
Your mother is a nosy, intrusive, interfering & controlling old trout & she launched her totally unnecessary bullshit on you before you were even home from hospital. 
I am also annoyed that she thought it was an appropriate thing to say and I feel like it has ruined our relationship.
Of course she thinks it's appropriate. She is a nosy, intrusive, interfering & controlling old trout, who reckons she gets to have a say over your choice of name.
As to ruining the relationship - reframe this, & buck up, because her nonsense is NOT worth getting upset about.,
Her behaviour can hardly be a surprise - you already knew she is a nosy, intrusive, interfering & controlling old trout. She's just being true to form, & one day you might even laugh about it. I doubt it will ever be laughing WITH her, as nosy, intrusive, interfering & controlling old trouts tend not to have a sense of humour about their overblown sense of entitlement, but you should certainly be laughing at her behind her back. Go on OP - it's the best medicine.
When she brings it up again (you know she will) use the experience as a launch pad into your new life as your baby's Number One advocate.
"Mum - you seem to have forgotten you have no say in this whatsoever, so do me a favour & give it a fucking rest, ok?"
Let her be offended, (you know she will) let her act out (you know she will) & let your resistance to her be a line in the sand.
Because if you let her keep harping on about the name, she will be harping on about everything, & she will be making constant digs about your parenting, because those are some easy 'hot buttons' for her to press when she wants to undermine you.
I don't know WTF she was playing at, letting herself into your home & rifling through your things, & I strongly advise you to put a stop to it NOW. Or you will never hear the end of it, while you are on mat leave & she wants to come round & tell you how you are doing everything all wrong (you know she will).
If you can't face asking her for your keys back, just change the locks, invent a bland excuse about losing your keys/updating for insurance, & 'forget' to supply her with a new key.
If she pushes you on that - tell her to fuck off.
You are a grown woman, you don't need her ordering you about, entering your home without an invitation, going through your stuff, & constantly undermining you (I just know she does ...)