I am thinking of running both the basin mixer and bath shower/mixer
cold taps off of the mains, whilst still having the hot from the hot
cylinder. (Hot will not be pumped).

This won't work. Assuming the shower mixer is pressure balancing with
sufficient range, it will throttle off the cold mains to match the crap hot
pressure, giving you a pathetic shower. If it isn't pressure balancing (i.e.
manual valve), you'll get cold out of the shower head and the cold mains
will push back up through the hot water cylinder and dump its contents
through your overflow.

Your choices:

1. Use a venturi type shower mixer.
2. Pump the hot
3. Add gravity cold supply and pump hot and cold.
4. Install mains pressure hot water.

Options 1 & 2 will provide good results at minimum cost. 3 is really not
worth it. Lots of work for no real advantage over 2. 4 would be much more
expensive still, involving the replacement of either your boiler or hot
water cylinder and should only really be considered if either need
replacement already.

Christian.