Private tooth extraction in a "dental desert"

I'm Theo, I'm 21 and live in a "dental desert" in Wales. For 5 years I've had repeated infections due to an impacted and now-decaying wisdom tooth. No NHS dentist has had the capacity to treat this beyond courses of antibiotics, despite me being recommended to have the tooth extracted as soon as possible. My local dentist practice has an NHS waitlist of 5 years for a routine appointment, with further wait times for surgical referrals/treatments. These timeframes are similar to those in my hometown in England. Without turning to private dentistry, my only option is to wait for an infection to develop into sepsis (which I have been told is a matter of time), at which point the NHS would likely extract the tooth. Of course, I'd rather see a private dentist than risk my health.

It would still take several months for treatment to go ahead privately, with costs around £60-100 for a consultation. Surgical extraction can be as expensive as £1550 per tooth, though most I have found sit around £350-500 per tooth. These aren't feasible costs for me as a university student from a single-income family, so I'm asking for help to finally get this tooth out before it decays my other teeth more or makes me seriously unwell.