The issue is that the main private lab in London (TDL) is swamped and almost all of the private clinics thus have turnaround time issues. I used Doctorcall (branches in Harley Street and London Wall) this Monday morning with a doctor taking the swab and got the results on Wednesday morning - the lab report from TDL was dated Tuesday evening. They said they do have some 48 hour urgent lab capacity but had none left on the day I went. The service was very good and the price (£175) was okay given that everywhere else I tried at short notice was unavailable. They said they are changing lab next week (presumably time try to get more reliable turnaround times).
I have also used Citydoc but they are no longer offering the nose and throat swab.
Beware there are some more entrepreneurial outfits advertising on Facebook and using Korean tests in house rather than sending to an accredited lab.