Just revisiting this thread for a bit of closure. As it happened, my Dad took my bike to the LBS the next day, got it fixed, and made friends with the proprietor. He was sold a Tyre Glider; and after picking my bike up (and sourcing a pair of cheap rubber gardening gloves) I cleared my mind of preconceptions and had another go.
Getting the tyre *off* was much easier with the Tyre Glider. I did have a good go "freehand" but didn't really get anywhere. However, putting the tyre *back on* was an absolute revelation. I managed it entirely freehand, and whilst it wasn't easy, I never got close to swearing etc. -- the gradual approach using your palms with the wheel on your knee really does work!! (together with an appreciation of the use of the "lip" vs the deeper part of the rim)
I genuinely had a preconception that tyres "ought to" just pop on and off, given enough oomph with a set of levers. I don't know why, but at least I have now seen the light!! Thanks everyone for your advice.