There is no formal qualification for an electrician, he only needs to be able to do the job to call himself an electrician.
For domestic we do have some laws in regard to how work needs notifying to the authorities, which varies around the UK, since your in London which is in England the English rules apply, so most work does not require notifying, think from memory as I live in Wales, it is bathrooms and consumer units only.
The notifying can be done in one of three ways, either the electrician is a scheme member and can self notify, or he needs to get a third party inspector or the LABC to inspect his work. Seems daft but it is up to the home owner not the electrician to ensure either he is a scheme member or some other inspector has been engaged BEFORE THE WORK STARTS.
As a result there are very strict rules as to showing the logos of scheme providers, if the van he roles up in had the name of the scheme provider on the side, or paper work shows name of scheme provider, then the responsibility for registering the work becomes the electricians.
So all rule book says is "A person who possesses sufficient technical knowledge, relevant practical skills and experience for the nature of the electrical work undertaken and is able at all times to prevent danger and, where appropriate, injury to him/herself and others." nothing about have 18th edition, or inspection and testing exams.
When I did my apprenticeship I did go to collage, but there was nothing at that time to say I had to go to collage, throughout my working life I have been required by employers to attend many lessons to be updated, be them sitting on tool box called tool box talks, or returning to collage, but nothing says they must send me, all they need to do is convince the heath and safety executive they have taken reasonable steps.
The Part P building regulations came about to fill the gap where other laws did not cover, plus make it easier to take to court. The fees charged by LABC effectively stopped legal DIY, oddly DIY still allowed with gas, only with electrical was DIY stopped. But doing any work for payment then electricity at work act applies anyway.
There are some areas controlled by the distribution network operator (DNO) and in theory only some one authorised by the DNO can work on them, the DNO does not follow the BS7671 document they have their own book, there seems to be a grey area as to if the DNO or the user needs to provide a method of isolation, the HSE state the electricity must be isolated else where, so turning off the isolator on the consumer unit is not good enough to do work on the consumer unit, as least in theory, so to comply with HSE it could be an electrician has not option but to draw the DNO fuse, however he clearly does have an option as to replacing the fuse, and he should not replace it.
The same with Part P, it is permitted for an electrician to do emergency work and inform the LABC after the event, however now moving into let the courts decide which is clearly rather unsatisfactory we really need to know before if allowed.
I have gone to an interview armed with my certificates, and showed them at an interview, but never have I taken them to a job, only thing often required is proof of insurance. And for sole traders often firms ask for silly amounts of insurance cover, but an electrician, plumber (not gas safe), builder, or any other trade traditionally had guilds, or unions, and the guilds and unions could and did run closed shops, but in the Thatcher time the closed shop was stopped.
The scheme providers are not allowed to share information about people in their schemes and they has to be more than one scheme or it would break the law Thatcher made. So an electrician thrown out of one scheme can simply join another.