Wolf... if you take the immersion out, then, as you say, you'll need hot water supplied to the kitchen sink and the bathroom basin.
You could fit undersink water heaters to each of them.
But.... you will need wiring for them which, for the basin in the bathroom gets complicated to meet the regulations.
It can be done though.
Alternatively... fit a small "Multipoint" in the airing cupboard in place of the immersion cylinder.
Unfortunately, they tend not to give a very hefty flow. Nothing like the flow from a tap.
Also, being around 10kw, it would need a new 10mm cable from the consumer unit to the multipoint. That can be an expensive faff if there's no direct route.
The plumbing would be easy though. Just connect to the existing pipework.
All in all, I really think you would be just as well off keeping the existing cylinder