I was told by an NICEIC approved electrician that by law an electrician can not decline to give a certificate for work he has done.
He has to give certificates for work completed, but since not complete then no he will not have to issue a certificate, although called a compliance certificate when issued by a scheme provider it is actually called a completion certificate when issued by the LABC.
The same happened to me well my son shouted at the builder for not completing on time which for our job was really important as needed to be complete before my mother left hospital, and they walked off site. I then approached the LABC to ask what was required to take over the job, at which point I found the builders had failed to inform the LABC that the work was to be done. Lucky for me is was for my mothers disability so there were no charges.
However you may be in the same position. You will clearly need to inform the LABC that some one else is going to be doing the work, and you can either get an electrician who is willing to take over the electrical work and issue a compliance certificate for whole job. Or if you can't find an electrician willing to sign off some one else's work then it needs to be done through the LABC and a triple signature installation certificate is used where in theroy the new electrician signs for what he has done and the old one signs for what he has done. In my case the certificate was endorsed with limitations and the design signature was left blank, it seems the LABC was only really interested in the inspection and testing signature and issued a completion certificate on the strength of that. They also it seems did some thing which resulted in the builder to stop trading. Which in turn resulted in us not being able to get any money back.
You need a completion certificate from the council, and any electrician will realise that, this is for the building work not the electrical work, however it does mean the council is involved so there could be a problem if the electrician does not do things by the book, in the main the council have there favourite electricians, I know they shouldn't but in real life they do, these are people they trust so they are not worried about signing off work they have done. It may be helpful to talk to council first and see what they say. It is not as if you can not tell them, the building is already started so they will already be involved.
The compliance or completion certificate are legally required. The installation certificate is not legally required, but in real terms it is required in order to get one of the legal documents. A scheme member electrician signs a legally binding document with his scheme provider to say he will follow the requirements of BS 7671 which in turn requires he issues either a minor works certificate or an installation certificate for all work COMPLETED but this is with his scheme provider not you so only the scheme provider could take him to court for not issuing one.
With a installation certificate the three signatures are for design, installation and inspection and testing, what could be in depute is who signs for installation as the electrician one has installed the cables, but electrician two will install the sockets etc. Much will depend on the stage of the other building work, if you need another builder to complete it then that builder may have an electrician he uses and he may do all the work required to keep the LABC sweet.
We were rather dismayed when we involved the LABC who in no uncertain terms told us it was the owners responsibility to inform them of any work being undertaken and although builders would normally do it for the owners it was still up to the owners to see it was done. If the electrician either with sign writing on the van, or advert in papers or any other means lets it be known they are scheme members then the responsibility shifts for electrical work to the electrical firm to inform the LABC through the scheme provider when work is complete.