Anyone in England and Wales can undertake an electrical installation condition report (EICR) which is in practice the inspection and testing of an installation done by some one else. As to what the HSE would say if the person doing the report was not qualified or insured is some thing else, but it would only happen if some thing goes wrong, in real terms in a private house some one needs to die before they get involved.
Scotland has slightly different rules, with rented property some qualification is require, but I don't really know exactly what is required.
Where the work requires notifying the LABC can decide if they will allow some one to certify the work, and it seems the government has no idea of wording used for many years, with competent meaning the person can look after the safety of others as well as themselves and the IET/Bsi have now removed that word from their book to try to avoid confusion, certify is not so easy to remove from regulations, I do wonder if the whole idea was to word it so courts would need to decide at a latter date what it really means.
In this country the Unions had some thing called a closed shop which allowed them to police to some extent who could call themselves electricians, however when this was made illegal, their ability to police was removed, so the Part P and the scheme operators replaces the unions, the scheme operators do in practice run a closed shop, but get away with it because in theory you can use LABC.
In practice the LABC is so expensive it is not really an option for small jobs. The Part P law says you don't need to notify but you do need to follow one of the European regulations, it does not need to be BS7671 you could for example use the German regulations however I would find that hard since I don't read German. But in essence it says it needs to be safe.
Remember Part L and other Parts of the building regulations must also be followed, they are just as bad as Part P to work out what you can do, so in real terms use common sense.
