It is not illegal for him to do work without issuing paper work. Unless in his advert states he is a member of a scheme. Then he must issue paperwork. It could be claimed not issuing paperwork shows he does not have a warranty of skill, but although he would have broken regulations he has not broke the law.
There are two ways to deal with the problem of no paperwork, one is to get an electrical installation condition report and hope any buyer will accept that, the other is to tell the local authority building control what has happened and pay their fees and if all OK they will issue a completion certificate, I would go for former as latter is likely to be expensive.
Although you can wire a house with surface wiring in a week with one person, in the main we sink cables and two weeks for two men is a more reasonable time scale. Since it has been completed so fast, it would to my mind ring alarm bells, pictures on the forum will help.
There is nothing to stop an electrician from using the LABC rather than be a scheme member, it would cost more, but he or you can pay the LABC fees, and he can either test it himself and forward paperwork to LABC or the LABC can arrange for themselves or a third party to inspect and test, for an electrician moving into the domestic from the industrial side of the trade, he would likely need to do this to be accepted onto a scheme. The problem is it is the owner who is responsible to ensure this is done. The electrician would normally do it for you, but it's up to you to ensure it's done, or use a scheme member electrician.
As a result jumping in with both feet and telling the LABC what has happened could cause you a lot of problems, you can't untell them. If he has intended to use a friend or other not quite correct method to get paperwork, involving the LABC could mean he could no longer do it. I was lucky, the work I had done was for my mothers disability, as a result there were no LABC charges, however the firm doing the work closed down, we think they were forced to close due to the LABC leaning on them. So we could not get any money back for work required to rectify their work.
We were also lucky to have friends who did work we could not do our selves, and the LABC accepted my signature on paperwork when we took over the job, in hind sight I wished I had kept my mouth shut, we felt we had done nothing wrong, we had selected a firm off the internet who were local, phoned and met up with them, looked at pictures of their work in the past, it all seemed great. Had they not bought a load of out of date tile cement which would not set, they may have got away with it. However that cement was their down fall they could not complete in time, so my son and I got involved, we had left it to my sister and father until things went wrong.
The guy from the LABC was very forceful when he told us it was up to us to register work, not the builder, only exception is where the electrician is a scheme member. So that flyer is very important, if anywhere it says he is a scheme member then your in clear, you phone his scheme provider, otherwise, be careful, is you tell council it could cost you another £500 just to get the paperwork.
I hope he has just gone on holiday and on return all is sorted, but I also hope my tail of woe will make you think twice about telling the council.