Success!



After posting here I had long think about what the problem was and once I identified it I managed to find a solution which worked.

I got the point about the air in the tube pushing the tire off, but generally I didn't put almost any air into the tube and when I was wrestling with the whole thing I had no tube in at all.

Problem: tire too stiff, and wrong shape. The bead was pushing out of the rim because the rubber did not want to remain in the curved shape required to stay on the wheel, the material retained the memory of its shape like a spring that returns to it original length after being stretched, that's why it kept popping out. The fact that the rubber is stiff meant that it did not remain in the position on the rim and it jumped back to its original shape.

Solution: stretch the tire and tuck the beads underneath the thread area, wrap it tightly with flat string into the tube shape it would have once on the wheel, with beads as close to each other as possible. Then basically walked on the inside of the tire in socks trying to increase the flexibility of the rubber and trying to reinforce the fold. Also went around it squeezing it in my fist into the required tube shape. The aim was to 'break' the rubber a bit, increase its flexibility and to manually mould it into the required shape. Then left it tied up for the night.
In the morning I managed to put it on without much trouble: it was still stiff and hard work, but it did not jump out of the rims. I did follow rpenmanparker's suggestion of inflating it after I've put most of the tire on.
I could not really apply the 'proper' technique because the tire was not flexible enough for that and I had to use my tire leaver to put the last bit on, but overall it was relatively easy. What a tire!
*Phew!*
(still the front wheel to change though)