Not really my area of expertise. But I will say I had a house in a very lime scale rich area of the country, the old fortic tank immersion heater went, and when I removed it I found the bottom of the tank was inches deep in limescale 'platelets'.
I dug it all out, fitted a new immersion, and all worked fine.
I'm assuming your plumber is inferring there's a build-up of limescale on the coil, rather than just in the bottom of the tank, because surely the latter could be rectified without changing the cylinder!
I'm not clear if your mystery lack of H/W is something that suddenly happened, or if it's been inherited. If it's happened suddenly, I don't think the limescale theory "holds water"🤪.
I'd be wary of changing things before understanding what is causing the problem.