The problem is most likely down in the valve assembly itself, Popple.
Once the head is removed (the bit you pull), there should be a way of dismantling it.
They're pretty simple. It's probably just a sliding bit that travels between two ports.... one for the taps, and one for the shower.
Dismantling may be from the top or underneath, where the hose connects.
It's a matter of figuring how to take it apart. Once you have, it's only a matter of greasing the rubber O-rings. Silicone grease, or even Vaseline will help it to slide more easily.
That's your challenge ... how to take the valve apart.