My experience of the private sector is that they're only interested in queue jumpers and the worried well, if you're looking for anything as contentious as a second opinion they don't want to know.
I had an account with Babylon a while back but it was a waste of money, every time I asked them anything they just told me to go to my (NHS) GP. It was the same when I wanted a CT scan reviewed, every private health company in the UK refused. More recently I'm trying to get an appointment with a neurologist, but again, I can't get one without an NHS referral and even if I do they won't review NHS MRI scans.
I had a second opinion via the NHS a decade ago, and
The trouble with that is that you then have a monopoly, in fact, the problem with healthcare in this country is that we already do have a monopoly for the majority who can't afford private. Even for those who can afford private, they're still dealing with the same NHS doctors. The conflict of interest in having HMG as the funder, provider, regulator, inspector, and policeman of healthcare ought to be obvious, but it apparently isn't.
Every time anyone tries to start a debate about how healthcare is run in this country they just get the American system thrown in their face, but nobody else in the world does healthcare like either the USA or the NHS, they're both odd ones out. There's no reason why healthcare has to be state owned and run in order to make it free at the point of use.
I'm in Canada at the moment. Health Care insurance funded, free at the point of use. Private Health care not allowed. GP claims from insurance direct, not patient. Hospitals self governing with hybrid state and province funding and claims to insurance for care.
D1 and D2 have both had excellent seemless service, including fully funded extensive physio (a key NHS failing)