Hi Redstar! I'm sure others will be along soon who are more experienced with the NHS, but since you and I are in similar situations, I'll offer my experience so far to help ease your mind...
I arrived on a spousal visa a few weeks ago and am now 29 weeks along. I had to go into my local GP's office (found via google and walking through our neighborhood, but I believe there is an NHS search tool online) and ask in person to register for the NHS. They gave me some paperwork to complete and I was required to bring proof of address and ID in order to register. I had already been added to our flat lease, so I used that as proof of address, but they told me they would also accept a utility bill. They keyed me into the system the next day, and I had appointments scheduled a few days after that. About a week after filling out initial paperwork, I got a letter in the mail with my NHS number. So, it was rather quick.
We're in an unusual situation in that we're moving in a few weeks and so I'm trying to register to deliver at a different hospital that is not near my current GP's office. FWIW, I was told I can "self refer" to the hospital where I want to deliver but I couldn't complete that process until I had my NHS number. Once we move, I'll need to register with a new GP in our new city as well. We're touring the maternity unit at the hospital I'm delivering at this weekend, and I'm hoping to complete the self-referral process then. I'll update this thread on how that goes.
Anyway, I suppose all of this is dependent on your local GP's office, but for me, it's been very quick so far. Hopefully same for you.