I'm pregnant and due in April, and at our hospital we're allowed two birth partners, so my DP will be one, and my mum will be the other. If DP has any issues with it he hasn't mentioned them, and neither has the rest of his family, and my mum is lovely - she's supportive without being interfering, will keep a cool head if DP freaks out and will absolutely understand if we ask her to leave the room at any point - she's not the sort to be pushing in for first cuddle etc 
So seems like there's no problems, but from reading some threads on here, I feel like I'm BU to DP for having my mum there too... is this likely to cause resentment in the future, either with DP or with members of his family about my mum being first grandparent to see the baby etc?