I had the same problem, a builder ran off, and he had employed an electrician and I did not even know the electricians name.
I had assumed the builder had done everything above board before running off, and so my first step was to go to the shire hall and explain I was going to take over the job, however it transpired the builder had not informed building control about the work to be done, and I was told in no uncertain terms it is the owners responsibility to ensure the LABC is told, the builder may do it for him, but still the owners responsibility.
Rest of work no real problems, it was just the electrics where the stumbling block was apparent. The LABC inspector was quite open, he did not have the skill to inspect the electrics, so at my expense he wanted to employ some one to inspect it. I did not want this, as was going to cost, and in the end he agreed to accept my signature, I do have C&G 2391 which says I have the skill.
However what he wanted to do was employ an electrician who would use the EICR forms, he would be told to treat it as a new installation, but as already said he can't write out an installation certificate for work he is not in control of, but the EICR is very similar to the installation certificate, except he is not signing to say he designed or installed the installation, he is only signing to say he has inspected and tested.
However the EICR can be completed with only a sample inspected, also it only looks for dangerous and potentially dangerous (C1 and C2) items, it does not point out all items not complying with BS 7671 valid at the time of the design. So the LABC have to engage the inspector with instructions that he must treat it as a new build, and he is not really doing a standard EICR, all he is doing it using the EICR forms, he is being far stricter with his testing.
So there is no point in you commissioning an EICR this needs to be commissioned by the LABC, you just pay for it.
In my case, when the LABC inspector visited the test equipment was laid out for him to see, complete with calibration certificates, and my as it was at that time C&G 2382 and C&G 2391, so he knew I had the skill, and clearly I could have written out either an installation certificate or an electrical installation condition report, but the question he had was can he trust me? So he originally wanted it inspected by some one he knew he could trust.
I had to convince him, I would not put my parents in danger, and would do a good job.
As a post script, when my parents died, and we came to sell the house, I initially could not find the paperwork, so I applied to the LABC for replacements, to be told it would take 4 months, and I would need to pay for the time it takes the council worker to find them, no fixed fee, in other words go away, it's just a tax, no one really wants to store these documents for you.
The solicitor said no problem, we just take out insurance, but then I found them.
He showed us the installation certificate for this house, and the readings did seem suspect, but I was not really worried, and after moving in I found they only covered about 1/3 of the house, lower floor only of a three story house.
Since I did not need a mortgage I don't know what lenders would say about lack of paperwork, I would have said a year or so ago no point in the seller getting an EICR done, however in a Pembroke EICR court case, the trading standards took an electrician to court for not reporting the defects to the seller in an EICR, when the buyer complained.
I tried to find the link, but only found how a Landlord was fined £1,600 for not providing an EICR, and banned from renting property, with such a low fine, he is likely not worried, and since he can't rent property it means the tenants have needed to find a new home, yes the home will be now brought up to scratch, but does not help the tenants much.