it can be done by a plumber or by an electrician.
If the existing electrical installation is very nicely done, with a 40A or greater MCB, an RCD protected circuit, a 10mm cable, and supplementary bonding in the bathroom supported by an electrical installation certificate, just ask a recommended local plumber to fit a new replacement electric shower which is within the Amps rating of your circuit (it is not usually worth repairing, unless you have a fine expensive one.
however, if the electrical installation is old or ramshackle, get a recommended local electrician to do it and to bring you up to date. When phoning, and before making an appointment for him/her to inspect the job and give you a price, ask which Competent Person Scheme s/he is a member of, and how long s/he has been a member (you can check on their website). Do not call anyone with a non-geographic or an 0800 number, or who only has a mobile. Do not use the Yellow Pages. Get recommendations from friends and neighbours. You want a recognised local landline. Don't give your postcode or full address straight away, mention a well-known local landmark and say "I live in Acacia Avenue just down the road from St Martin's church" or "Nelson Mandela House next to the Tesco Roundabout" This will frustrate national clearing agencies who pretend to be local companies but will sell you as a prospective customer to any tradesman who has no customers or reputation and is willing to pay them to get work.
You haven't said what is wrong with the old shower