If you wanted to see a physio in Coventry, although he's not a climber I can recommend Matt Hardy at the Warwick Uni Physiotherapy Clinic. I think he only works there on Tuesdays, but I was able to get evening appointments with him and he was a great help in rehabbing a (non-shoulder) injury I had earlier this year: https://warwick.ac.uk/services/sport/health-and-wellbeing/physiotherapy/