The system is S plan and if I turn the pump speed up, I get pump over.

What is the boiler? You might want to consider switching to sealed
pressurised operation, which eliminates pump over and allows higher pump
speeds.

So the question is - should I go plastic for the larger 22mm pipe and
replace existing copper?

Probably. You may find you need to fix the pumping over at high pump speeds
even then. However, 15mm does sound very marginal for the load and 22mm is
probably more realistic.

If so, I understand that this shouldn't go right up to the boiler. If not
how close?

2-3m or so, although you should specifically check the instructions of both
the boiler and the pipe manufacturer and use the largest figure you find in
either. It's more of an issue on older non-condensing boilers.

Christian.