I apologize in advance if this will come off as scary, but I don't want to paint a rosy picture that's far from reality. That being said, there are lots of variables with NHS care (e.g. where you live), so your mileage may vary - you may have a very positive experience with NHS mental health services.
The fundamental problem is that mental health services are one of the worst funded specialities in the NHS at the moment. The NHS does cover mental health services (therapy sessions, medication, etc.) but timely access is typically problematic unless you are acutely severely unwell.
Most of the time, GPs are able to prescribe medication you are already taking, so if you have a regime that works for you, bring your medical record and a signed official note from your current doctor detailing your treatment and diagnoses. Be specific - the more comprehensive and detailed your medical history is, the more likely you'll be able to get your medicines switched over and not have to potentially start from the beginning to get diagnosed (again) to get your prescriptions approved.
If you are able, also try and bring over a large supply of medicine to keep you going until you get your prescription sorted with your GP. It may take some time to get everything sorted and get your foot through the door: you'll have to get a piece of mail with your address on it to prove residence for the GP catchment area, then you'll need to register with the GP, you'll likely need to book a new patient consultation (which may not happen for a few weeks), and THEN you could discuss setting up your prescriptions. If this isn't possible, when you register with the GP, insist that you need a new patient appointment as soon as possible due to your medication supplies running out.
I don't think that getting medicine will be an overly laborious hurdle once you get registered and have your first appointment, but access to a therapist could be the challenge.
Generally speaking, therapy referrals aren't quick unless you are an immediate danger to yourself or others. I wouldn't be surprised at a 14-16 week wait for a first session. Some councils have basic talking therapy sessions with a slightly shorter wait time (and in these cases, you can often self-refer), but if you're after a proper mental health professional, you will likely be waiting for a while or paying out of pocket for private sessions. NHS sessions (once you get them) are typically every 2 weeks, and you only get a limited amount of them (8-10) at one time before a review to see if you still would benefit from them.
The good is that the mental health care I've received and my friends/family have received is fine. None of us, in our experiences, felt like the care was sub-optimal. And while in a dire state now, there is a growing push to try and get more funds allotted to mental health services and more recognition that mental health is serious, and must be addressed on equal grounds as physical health.
So, I guess the TLDR version is: the NHS will cover mental health care, the actual care will likely be fine, your GP will probably be able to prescribe your medication, but the wait for therapy will probably be significant.