I am very lucky that my hospital has strict visiting hours: 6PM-8PM, with siblings being allowed to visit an hour early.
Are you delivering at the same hospital as your SIL? If not, I suggest looking into your hospital's policies. Maybe that'll solve your problem.
If not, I would either:
A) warn all family members that you and your H wish to spend those first few moments/hours alone, as family of 3 - See with the hospital staff if they will help with this request (aka: if they will turn away people who would decide to show up anyways)
B) Wait until you're ready for visitors before telling anyone the baby is there. You can always say you got so caught up in the moment your forgot to call (but your fam might call you out on it, so I would probably not recommend it)
C) Do both A and B. And if anyone says anything, you tell them you had already told them about your wish to spend that time alone, so there was no point in calling them early anyways.
Also, I think I might give my own family a head start, just to make sure your FIL's gf is NOT the first one to hold the baby. Just because I can be a b!tch that way.