A general rule of thumb:
For a brand new installation - it's an electrician's job.
For maintenance and replacement of plant ie valves, pumps, immersion heaters, cylinders, its a plumber's / heating engineer's job
For fault finding, if something's not working as it should, your best bet is to use an electrician to narrow the problem down, then a plumber to change the part (if its part of the plant), if required.
Hot water and heating controls are generally a bit of a grey area between the two distinctive disciplines and requires a tradesman who specialises in one, but has good experience in the other, to carry out these works.