I am also based in hte UK and have had shoulder problems in the past. Without private health cover it could all take a long time as you say and considering the cost of MRI scans, the GP will avoid and dealy sending you for one as long as possible. Unless it is impacting your normal daily life outside of sport they are unlikely to be interested in spending that budget. Hopefully you have an understanding GP who agrees that tennis is important in keeping you healthy.
In terms of the understanding of shoulder injuries I would not expect much int he way of certainty from your GP. I say this with great respect as I am the son of one and as my wife is also a doctor, we have many GP friends. The shoulder is very difficult to diagnose and even the specialists don't find the answers easy.
My story last year involved a visit to the GP to whom I explained my shoulder injury. He asked me to raise it to the side and front, asked where it hurt, how I did it etc. As it only impacted my tennis or ball throwing abilities there was almost no interest. Fortunately they were willing to write me a letter for referral as I have private healthcare. I was then seen by a sports specialist doctor who is now a professor in a shoulder clinic at a private hospital. After unltrasound, examination and MRI, the best guess was SLAP tear, slight impingement and bursitis as a result. There was no certainty despite the MRI and the only way to tell was to let the surgeon scope it with a view to repairing what they were expecting to find or not if their best guess was wrong. In the end they were right and it was repaired and it has all healed over time.
I am lucky as I have private cover, but if you don't and don't fancy running up large bills paying directly, then you will need to make sure you impress on your GP the impact it is having overall. Hopefully they will be helpful, but the NHS is not well set up for sport injuries in many places although there are some specialist clinics I know of in London hospitals. Make sure you research what is nearby if you want to push for a referral.
Best of luck with the injury