It is permitted to DIY a re-wire, in England two ways, in Wales only one. So in both countries you can apply the the LABC and they can allow you to do rewiring which they deem you have the ability to do, and either they will check it, or they will appoint some one else to check it. In England some scheme providers can allow some members to do third party inspection and testing as well.
Both cases you do what they permit, and they do not provide you with an installation certificate, the only person who can complete the installation certificate is the person doing the design, or doing the installation, or doing the inspection and testing, you can have three signatures, but that means only you can fill it in.
The LABC if they feel you have the skill can allow you to complete the installation certificate and submit it to them, and they can test selected items to see if their results match yours.
I have done it, and they wanted to see my test equipment to show I had it and also my qualifications, in my case a degree in electrical and electronic engineering swung it, and I had to submit the installation certificate and then they issued a completion certificate on the strength of it.
The problem is getting permission to turn on a circuit, it is up to the LABC inspector as to if he permits you to re-energise the circuit without it being inspected or not, and if he inspects or if he uses a third party inspector.
So you could be allowed to do it all and then he gives it a quick look. Or he could ask for each section to be tested before it is re-energised, a little thought and you can see the problem, either you end up having to do the whole job with no power until complete, or you have to pay for some one to inspect as every bit is completed.
Typically to inspect and test people charge by the day or half day, at just £150 as day, that is £75 per inspections, so 6 rooms in a house, that's £900 plus the £100 all plus vat for the LABC, this is where it all starts getting a bit silly, mother house rewired for £2500 so to break even you need to rewire for £1500, which likely you can do, but all the hassle to save £750 is it really worth it?
So you have to find out what your LABC inspector will allow, my son did work in Liverpool and they were really helpful, Cheshire were not quite as easy to work with, but not bad, Flintshire was a real pain in the neck, so there is no universal answer, it is down to your inspector, but clearly you need to have the skill, and the test equipment before you have a chance.
So I would say you with need some paper to show skill, C&G 2391 for example. I know you don't need the paper in theory but how else are you going to show you have the skill.
I did it around 2006, I had C&G 2382 and 2391 but the inspector was not satisfied, it was the degree which swung it.