At present we need both the immersion and the boiler to give us a decent
bathful of hot water.

Do you mean that your cylinder is heated from the boiler anyway, and that
you use the immersion as well to get that little bit of extra temperature to
fill the bath before it goes cold?

If so, the problem is that your cylinder is simply undersized for your
purpose and should be replaced by a bigger model. If replacing and your
mains pressure and flow is good, you may want to consider replacing with a
heat bank or unvented cylinder, which will fill baths in no time at all and
provide a superlative shower.

Christian.