Its not just going to be 500 quid - you'll likely need to go see an Orthopod first for a consultation, get him/her to send you for an MRI then have a followup appointment with them...
consultation fees could be 150+ for each of those appointments
and then what do you want - presuming the MRI shows something that could require treatment are you going to pay in cash for that too?
I'm not sure you can mix and match here - AFAIK either you use the NHS or you don't, don't think you can't just pay for an MRI/consultation and then ask your NHS GP to refer you to an NHS consultant and bring with you the MRI/diagnosis you paid for privately
The only way it would be fair is if the NHS only allow the use of private scans if the patient is then made to wait the same amount of time they would have had to wait if the scan was done on the NHS!
Otherwise people with money are cue jumping which is not cool or fair and completely against the whole FREE health care ethos imho.
Your physio most likely change their practice due to the scans and therefore the only reason you seem to want this scan done seems is for litigious reasons. In which case the amount you will get most likely won't really be that significant when you are yourself admitting the damage will have healed in such a short time frame.
If your physio is saying it will guide her treatment then it's her responsibility to request and refer not yours. If she is not allowed to request an MRI then she needs to request that the person who referred you to her does so.
As to your second part I would not be totally sure an MRI would give you the answer you are necessarily looking for.
I would be suspicious of any practitioner I was referred to from a solicitor. Not that this is anything against solicitors at all they are rather good at what they do and what they are trained to do. However, they are not trained to determine what the best course of action is for a client's health.
The benefits of not going to Harley St! Hope it gives you that answers you are after.