Private care in the UK can be very fast - e.g. I went from an initial consultation with a doctor for "knee pain" to having two sessions with a consultant and an MRI complete and booked in for physiotherapy within 48 hours. Things are often done the same day or within 24-48 hour turnarounds, although sometimes you can only get seen by specific consultants on certain days (e.g. they only see people on wed + Fri etc) . I am not sure what the on-ramp for that is as I go through my work health insurance but presumably you can just pay out of pocket.
www.hcahealthcare.co.uk seems to be what everything my healthcare insurance (from a BigCo) goes through, so could be worth starting there if you fancy a visit to the UK.
Good luck.