Home DIY

I need to speak to some boiler people and see if one of those new-fangled pressured water heaters will work in my place. If I had adequate mains pressure and installed a pressurized hot water unit, does it remove the need for a cold water storage tank as well as my current immersion heater?

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?