I can't comment on the procedural advice given here (but if I were in your position this is definitely one of the forums I'd go to for help as I advised you on MSE) but I would make a couple of general observations:
1. You say you aren't sure you've actually been overpaid. I understand that but I'd have to say from my own experience (and I've dealt with NHS overpayments from both sides of the fence) I think it's highly unlikely that the trust would be suing you if you had not been overpaid. So I think it would be prudent to assume that there is an overpayment. (Having said that, it could be a complete cockup by the trust and they just might be wrong.)
2. I know BankFodder has suggested that estoppel might be a possible defence but, again in my experience, it's rarely successful, but (a) see 3. below and (b) other posters on here may have experience of estoppel succeeding.
3. Salary overpayments in the NHS are extremely common and happen all the time. All the trusts I have been connected with have very well-tested procedures for preventing them in the first place or dealing with them when they do happen. As I said on your other thread on MSE, I'm amazed that it took them over a year (is that right?) to identify that you'd been overpaid (although that doesn't necessarily give you a defence in itself - they had six years to chase you).
When you handed in your notice, your manager should have completed an Employee Termination form (a form P4 where I worked) notifying payroll and HR of your leaving date. It sounds as if this was not done in time (or got lost) and was not actioned to remove you from payroll. But again, I don't understand why it took over a year to discover the overpayment.
To return to estoppel, I wonder whether these failures by the trust together with your change of bank accounts etc etc around the time of the overpayment might strengthen an estoppel defence by you and help to establish that you acted in good faith and could not be expected to notice the overpayment. I don't know.
4. A SAR is a good idea I think. I have no experience of them but I think I would be especially asking the trust for all information surrounding your resignation, why the overpayment happened, and details of how the overpayment has been calculated, and what they've done to chase it up. (If I were you I would be very interested to see a copy of your resignation letter and the associated Employee Termination form. I'd want to see if the resignation letter was date stamped when received and also the dates on the termination form). You'd also want to see any internal communications discussing the overpayment.
Whether you can (or should) ask for these in a SAR I don't know - you'll have to rely on more knowledgeable people here to advise on that.
Two possible complications re a SAR: first, I don't know how the timescales for a SAR tie in with your need to file a defence within a particular time frame.
Second, your old trust's payroll may very well be outsourced to a third party payroll provider and they may hold most of the relevant information concerning the overpayment - especially the calculation. (Maybe that's why the trust has not been helpful in this respect - they don't have the info?). I simply don't know if a SAR to the trust would cover this or whether you'd need to do another SAR to the payroll provider. Hopefully others here with more practical experience can advise.
I don't know if this is good advice or not, but I'd contact the trust again on Monday and ask for details of the overpayment and if they won't provide it, ask why not(?). I'd also ask who their payroll provider is so you can SAR them if necessary.
I wish you good luck because you are a bit of a victim here, but I don't think your chances are great. Sorry. (See what others advise - that's why I suggested coming here!)
EDIT: Cross posted with other posts but think my points are still valid. Also re the Termination form, the trust ought, if I remember correctly, hold a copy on your personal file as well as in payroll and possibly HR.