Dont assume its a rotator cuff injury.
If you can't wait to see NHS physio pay to see a private one. If no improvement over 3/4 treatments and exercises if given then they should refer you for an MRI scan.
The reason why I say this is I had a pain in my shoulder and thought it was rotator cuff, Physio didn't work went for a scan found out 4 things wrong with my shoulder and none of them was a rotator cuff issue because I go to the Gym.
So i need to have a Op to fix all the issues.
Dazzler