I've been seen by a NHS meno clinic. First referral I waited about 3 months.
Didn't find them that helpful - specialist told me 'menopause is not a disease' and when I expressed concerns that dose might be too low, they just told me let's wait and see as I'd only been on it a few months (which is probably the standard approach).
Saw a different one recently after things went haywire, much more helpful. I think being specialists, they have more knowledge about different treatment options, possible causes of problems, and also of broader issues that relate (e.g. PMS/PMDD) than GPs. They can also prescribe things that GPs can't or are reluctant to prescribe.
Access is an issue though. Too many patients, not enough doctors/clinics. After my initial appointment, follow up was scheduled 6 months later. Meanwhile my hormones took a nosedive, but was told no appointments available for several months. Eventually they did find me one though after a few weeks of repeated contact.
If waiting times are too long and you can afford it, I would consider private. Your hormones are so important to your physical and mental health and if you can afford a few hundred pounds for a private appointment here and there, it's probably worth it. For a working person it's probably the cost of a summer holiday, and I think it's worth giving that up for one summer if it means you get sanity and wellbeing for the rest of the year.Once you are on a regime, it's also possible that your GP can carry on prescribing what the private one did.