One of the best things about the UK — in my opinion — is that as part of the taxes you’ll pay, you have access to free health care. It doesn’t cover everything, and it isn’t perfect, but my goodness is it better than what we have in the US.
Once you have a place to live, you should register with a GP (general practitioner). Most will limit who they allow to register based on how close by you live. This site will help.
They’ll want proof of address (everyone does — it’d be so much easier if they had an option for a non-driver’s license ID like they have in the US) and your passport, which they’ll just glance at. You’ll have to fill out an NHS form and then whatever forms the individual surgery (what we’d call a doctor’s office) requires. After that, you can make appointments.
If you don’t have a GP, or you haven’t even gotten a place to live yet, but you still need some medical care, you can go to a drop-in clinic. I developed a wicked cough just a couple weeks after moving here and got some great care at a drop-in center.
The key is to register basically as soon as you have an address, because you never know when you’ll get sick. You won’t want to go to a drop-in center and wait between 20 minutes and four hours when you feel horrible. And if you have a condition that requires prescription medication or durable medical equipment, you probably already know this, but it can be easy to get caught up with all the other bits of life and then you’ve been in the UK for a month and suddenly realize that the behind-the-counter allergy medicine you get in the US is prescription only in the UK. Not great.