Home DIY

It does, yes. You would need a pressurised storage cyclinder plus expansion tank and all the controls to go with it which includes safety pressure relief components. New installs have 15mm pipe throughout whereas yours may have 22m (3/4" in old money). 15mm is sufficient due to the higher pressures forcing the water through the smaller pipe.
The higher pressures may find weak spots in the old system though (the part fed off the tank).
Are you on economy 7 electric?

Would the press. storage cyl and expansion tank fit into the same space as the immersion heater? The main idea of installing this setup would be to free up some cupboard space by negating the need for cold water tank along with having proper pressure from the bath/shower fitment (the flow almost stops when you get it head height).

Economy 7, yes.

I'd taken photos of the current setup to get a quote with but they must be on the old phone.

Guess it would help if I knew the capacity of the current one. Any idea what's suitable for a 1br flat?