Hopefully i'm not too late on this but here is my experience.
I got here in Sept and I immediately got added to our tenancy/electric bill. Within days I received mail. I needed 2 pieces to open a bank account. First I joined my husband's account (he made his own and made his old one our joint) and I made my own account at an entirely different bank. (just a preference)
Once I had all of the above I signed up with the local GP. Easy. I've since gone to the doctor and didn't have to pay
a thing. It's unfathomable and amazing to visit the doctor without worrying about money.
After I had the mail/a bank account I was able to take the Tesco sim out of my phone and get a contract with 3. Now I have a permanent UK number. I'm taking driving lessons (just to refresh/get used to the roads here) but I am still in the process of getting my provisional-apparently it isn't difficult to get that but the test is hard so I'd recommend taking a couple lessons and ask them to train you "for the test."
I haven't gotten my NIN Yet, I'm just waiting until I have a job. I won't go into details but I've had a really, really difficult time even getting someone on the phone to make an appointment. If I get taxed a bit more for a few weeks, that's my problem. I've been chucking out applications for about 2 months and I've had 3 interviews, hoping to hear back from a possible position this week. I've been told that's fast to get replies-everyone's experience is different. I've seen people on here who waited 6mo-1year before getting interviews to people living here 2 weeks and already having interviews. Took me 2 months for someone to bite on my CV.
That being said, not sure what your plans are, but in regard to the NIN/working here/etc I would also recommend volunteering. I volunteer at the local museum. I also go out constantly and walk around town even if it's on my own. Not sure where you're living but if you can walk and explore, take advantage of that. When I first got here (literally 2 weeks after I'd arrived and had all my paperwork settled, BRP, etc) I joined a local Writer's Club, got a library club and basically attended local meetings on events, etc and I have already made a few friends of my own here. I know that isn't easy for everyone but if you're a people person you might be surprised how quickly you make friends/acquaintances. I was told English people are quieter/cold and not social but I find that to be a bit untrue, at least where I'm located. People talk to me just like they would at home! I've had no trouble meeting people. That being said, I'm not shy at all. It might not work for everyone.
I am having a really difficult time now because once our wedding passed in October (long story short we got legally married in the states in June, had a wedding in Rochester in October) I had nothing to look forward to and the "newness" of moving here wore off. Right now it's cold, gets dark at 3:30 and it's Christmas-in short I'm sad. I held onto home and everyone back home moved on with their lives. Moving here has been exhilarating, life changing and exciting. But also stressful, isolating, and it's a huge adjustment. If you ever need to chat during a good or a bad day, feel free to message me.
I hope you get everything sorted! Welcome to the UK