There is this guide for landlords in England Wales and Scotland are different. It simply says "Landlords must then provide written confirmation that the work has been carried out to their tenant and to the local authority within 28 days." for the electrician to do the inspection it states he needs insurance and a level 3 qualification, but I have seen nothing about anyone doing a repair, clearly if the repair will need Part P registration then either you need a scheme member or have to pay LABC to get a compliance or completion certificate, but personally I would say if you have before and after photos as proof, then as with all health and safety things, even a note on fag packet would do.
It does seem strange in one way, but I have had it where some one has phoned with a fault, I have gone to the building as I have heard "unit 5", but could not make out what the guy was saying, on getting there no one has admitted phoning and not fault found, and only thing I could do was walk away. It was found again latter and fixed, but it did point out how to be traceable it needs to be written down, however a text message is considered as in writing. So you can email the tenant or text them but can't simply talk to them on the phone. With most health and safety the writing down, is to avoid errors, and keep a record.
Where the problem arises is where an inspector reports a fault, but when an electrician gets there he can't see any fault, this has happened many times, I have walked into a hotel with an EICR and to be frank wondered if the report was for the same building.