It is down to your local authority building control, they can issue a completion certificate, however they can't issue an installation certificate, only you can issue that, where the problem arises is getting the LABC inspector to accept your installation certificate, they may accept an EICR done by some one they trust, or themselves, but it is totally at the discretion of the inspector.
Clearly step one is to complete the installation certificate, telling the inspector,
Unfortunatly no megger tester so its not been done. It needs a full test.
Basic tests have been done with a fluke multi meter to make sure nothings got the wrong polarity or any broken connections
is not going to impress an inspector, you have to make him believe you have the skill required, and if I was the inspector I would select some random circuits and test, then see if readings match yours.
I have once used the LABC to get the completion certificate, as with you job started before I contacted him, we had employed a building firm to install a wet room, we argued with them and they walked off site, since social services involved we decided it should be done correct, so I contacted the LABC and said we would be taking over the job.
It seems the building firm had not applied to LABC and we were told in no uncertain terms it is the house owner who must inform them, although most builders do it for the house owner, it is up to the owner to ensure it is done. At this point I wished I had not told them, however because the work was for my mothers disability there was no charge. He seemed to think the RSJ was OK, and drains were OK, he missed that the lintel for door way was only supported on one side. He also missed the spacing was wrong for loo, then came Part P, that was only bit I was not worried about.
So the three test meters were sitting there for him to see, he wanted us to get a firm in to do a PIR as it was called then, however since my dad, my son, and myself were all electricians we felt we should be able to write out our own installation certificate, my son was a sole trader at the time, so had insurance, so we wanted him to sign the certificate, however the LABC inspector did not seem to know what a C&G2391 was, and was unwilling to allow my son to sign the certificate, my son said well if you want us to get some one in he must be better qualified than us, and my dad has a degree, so with some reluctance he said he would allow me to inspect and test, and if the reading were within limits he would issue a completion certificate.
So I tested and inspected, and sent in the installation certificate and about a week latter I got a completion certificate in the post, the council did not even look at the job once finished.
Since then whole house was rewired, but I was in a hurry, so got a firm in to do it. Also had the kitchen re-wired before that, and looking at the two compliance certificates and the completion certificates there seems to be no cross reference to show which installation certificate they refer to, so in real terms how any house buyer would know I really don't know, but I suppose there must be a record some where!
The insulation resistance meter is cheap enough, I got one for around £35, however it does not do low ohms, the loop impedance and RCD tester however are more expensive, I have been surprised how many new RCD's fail, in the main it seems due to strain from cables, slack off cables and re-torque up and they have passed OK, but clearly they do need testing.
Although it says you can find the ELI at origin by enquiry, the measure the cable resistance, what you want is when the inspector uses his meter, for him to have same results as you, so really you do need a loop impedance meter.
Of course if you get an EICR done then they will record the readings, and you could then cheat and transfer all readings to your installation certificate, however that will likely mean having the EICR done twice, one to get your readings, and one for council to get readings. So likely same money to buy meters as to get EICR done, so you just have to buy or hire meters. I tried to hire meters, however it seems they need calibrating after every hire, so not cheap to hire.
I know mothers house was not re-wired perfect, I would not think any house is perfect, so an inspector will likely find fault, this can work out expensive, as then you need to pay for another inspection, it could be some thing really daft, like brown sleeving missing where blue wire used as line. And the inspector can demand that work is exposed for testing and inspection, I know I have used rods to thread cables under floor boards, however also know the cables should be clipped.
So your at the whim of the inspector, he may use common sense, or he may be a jobs worth guy. I was lucky mine when though OK.
As to having a CU changed and getting a certificate for that, well it's down to who buys the house, they may be conned, or they may realise.
However if you get it wrong, then who knows what will happen? I read the court cases with interest, it seems people who have made errors can get away scot free when they don't have training, so electricians mate doing inspection and testing and getting it wrong got away with it, it was his foreman who got the prison sentence, so if some one accepts your work, and you don't have some qualification to show you have the ability, then the person who accepts your work could find themselves in jail if you have made an error. So reverse the situation, would you sign your name to work which could result in jail if you don't need to?
So real terms, either tell the LABC and do what they ask, or get an EICR done and hope nothing goes wrong, the point is it may not even be your workmanship which is wrong, with the foreman who ended up in jail, a plasterer had put a nail through cable, a plumber had not glued the pipe on a tun dish, and an immersion thermostat did not have a cut out fitted, and the boy friend told the woman to turn water off before the electric, it took all these faults to kill her, but court decided it was the foreman's fault for using unskilled labour to press a button on a test meter and rely on him putting down actual reading and not fudge up some results.