An update on this: I spoke to the builder. He was surprised it was notifiable but said not to worry, he would get the electrician to notify through his scheme. He said there was nothing for me to do, the scheme would tell the council and they would send me the relevant docs. This seemed fine.
I sent him a note once he finished on Friday to ask how long it would take to get the docs from the Council, and also to say that I was a bit surprised that I didn't have some sort of certificate to evidence the work and testing done. He said not to worry, the electrician would send him the certificate and he would then send it on to me - no need to tell the council, the certificate is just for my purposes when I sell the house.
So, I think it's all in order. Everything he says accords with my understanding of how these things work. The only slight concern I have is the "no need to tell the Council" bit. If I'm just going to get a minor works certificate then I'm still in trouble... but not as much trouble as the builder, I believe? I've had a look on the Planning Portal and it seems that it is the person who does the work, rather than the home-owner, who is liable to pay the fine for failure to comply. I guess it's not ideal for me either as I will still need to do something to ensure the work is properly signed off, but clearly he's running a bigger financial risk than I am if he's planning on doing anything other than having his electrician notify the council through the competent person scheme, right?