None really. The private services (really just a bunch of loosely connected freelancers) exist only to rubber stamp a gender dysphoria/gender incongruence diagnosis or write letters okaying them for surgery and then refer on to associated endocrinologist/private surgeon.
Any exploratory therapy or differential diagnosis is meant to occur before contacting the transition services doctors.
The most rigorous of the first stage doctors seem to be Dr Dimitri Popelyuk and his staff at The Gender Clinic (they at least insist on calling a family member or partner to corroborate the service user’s personal history story, which makes a lot of transition seekers angry) or perhaps GenderPlus ((headed by Aiden Kelly) who are at least CQC registered (genderists complain about them being expensive and requiring lots of unnecessary’ sessions, but GenderPlus are still gender ideologues from my vantage point, which is terfy stepmother to an teenage transman). Kelly is a psychologist and Popelyuk is a psychiatrist, iirc.
GenderGP is currently close to collapsing due to a combo of raising prices and announcing the use of AI in their systems, so basically charging more money for less personal attention.
I’ve seen very few stories of transition seekers being rejected or even overly questioned by private doctors - the psych side of things seem to weed some people out via online triage forms but once an appointment is made almost everyone seems to get a diagnosis (sometimes with two appointments instead of just one) and the endocrinologists only seem to pushback when service users have unreasonable requests regarding wanting to pick and choose between hormonal effects as though on the character creator stage of a video game.
Even some people who claim to be systems and use ‘we’ as a preferred pronoun have been waved through. The only person I’ve seen rejected by multiple gender services seems to want to be an early teenage boy forever (and in reality is an early 20s woman).
The only gender service that seems to insist on seeing people in person and is capable of doing blood tests in house is YourGP based in Scotland.
Gender Hormone Clinic (London) do seem to do a few sessions in person but that’s run by a nurse prescriber, not a doctor (it used to be part of London Transgender Clinic but the plastic surgeon who owned that has declared bankruptcy, and the hormone side was scythed off before the rest of the business went under).