As you were all so supportive, I thought I would let you know how the appointments went!
The scan went fine - fortunately they were not running too late so I didn't have to hang on to a full bladder for ages. It was not comfortable and predictably I did have leaks

but it was no problem at all as I had the necessary protection, and they were very caring and understanding - i was worried how they'd react to the fact I was wearing a diaper but I needn't have been at all. Fortunately, the scan showed my bladder does empty fully as it should.
The second appointment (seeing a consultant) was also much better and simpler than I had expected, and everything was done to keep as much dignity during the examination as possible. The outcome was that she thinks the problems sound like overactive bladder, and she is asking my GP to prescribe a medication for this. Apparently this one does not cause too many side effects - I'll wait and see how that goes. I am also going to have a pelvic ultrasound done to make sure there are no other problems.
If I get no results from the medication in a couple of months, then I may have to have a urodynamics test.
This was quite eye-opening in showing it is important to persist in looking for help for these kind of problems. I have raised the incontinence issues with different GPs over several years, but found it hard to press for investigations etc because I've always either been dismissed or told that it's part of another long-term health problem I have and that there's nothing that can be done. But the consultant I saw said that whilst it may be contributing, she thinks there are separate problems and that anyway this medication is likely to help. It's just frustrating that it has taken so long and for the incontinence to get substantial before I could get to this stage! Partly my own fault for not being more insistent though, I admit.
Even if in the end this medication doesn't help, I'll know there is at least some explanation of why the problems are happening, and that investigations will have been done to check if there are any other treatable problems going on.
I am so so thankful for the support on this board, because without it I probably wouldn't have dared to go through the process again to get help or to go to the appointments...

Thank you!!!