I'd go private if you can afford it.
I've pets and talking to vets (no insurance) versus talking to GPs is like night and day.
So often you can't have an intelligent conversation with a GP or even any specialist you may be lucky enough to see, because they're clock watching and operating on a quick-fix basis from a symptom perspective.
Bound/constrained by NHS prescribing rules and considerations of what constitutes normal (woefully inadequate. The NHS appears to have an acceptable suffering level which is purely based on outward appearances, so if you're the stoical type you're fucked).
Or when it's worth taking action (generally if you're holding down a job, you're "fine". If you're not working, then you're of no use anyway, so it doesn't matter if there's something else wrong with you and no point spending money figuring it out).
No interest in diagnosis (because then they'd have to follow NICE guidelines). No holistic approach to considering all symptoms or how other conditions already diagnosed might play into what's happening. No interest in how normal life (never mind chosen lifestyle) or well-being is affected. If your undiagnosed health issues makes you depressed you can have some antidepressants, they're cheap.
Whereas I can discuss my pets general health, mental well-being and athletic performance with a vet and if I say something is an issue and want it investigated, it's investigated. I can have a discussion about what investigations, likely outcome, cost and whether it's worth taking an educated guess and acting accordingly, without being treated like a nuisance for not blindly complying with the first thing offered or being offered nothing and told it's unnecessary.
When it comes to diagnosis I can discuss prognosis, treatment options, the merits or otherwise of taking a conservative wait and see approach, the pros and cons of gold standard treatment, the outcome I am (not them!) hoping for, as well as issues surrounding quality of life both for pet and myself. Coming to an agreement/decision on what we're going to do next.
The only guidelines they're bound by are the animal welfare ones. The only thing I'm constrained by is my finances and time available for nursing care.
If you've got complicated and seemingly random symptoms, you need someone you can work with, someone who's prepared to listen to you and take time to consider what you've said, someone who is there each step of the way so knows your full history. Not someone you've never met before, who doesn't know so much as your DOB and diagnoses, attempting to patch you up and send you back out there in a 10 min time slot because they're already overrunning by half hour and want to go home.