All of the schemes (NAPIT, ELECSA etc) only allow their members to notify work which they have done themselves. The certificate the householder gets confirms that the work complies with building regulations. It does not contain any specific details of the circuits, test results or anything else.
These are only required for notifable works, such as new circuits, work in kitchens, bathrooms, etc.
Not notifiying is against the law, and people could be prosecuted for not notifiyng LABC either directly or via one of the schemes. (Although it is very unlikely).
This is the same as if you failed to notify before fitting new windows in your house, or built an extension, changed the drains, installed a hot water cylinder and so on.
The other certificate (usually an Electrical Installation Certificate) is the one which contains test results and circuit details. This documents states that the installation complies with BS7671.
These should be issued for all work, and can be issued by people who are not members of any scheme (the only requirement being that the person completing the certificate is conpetent to do so.)
There is no legal requirement for an electrical installation to comply with BS7671, or to have a certificate issued.
If you want someone to test and inspect an installation to confirm it is safe, this can be done by any competent person. However, there is no legal requirement for this to be done, and people doing these don't have to belong to any scheme either. Additionally, no inspection will cover cables concealed in walls or floors, and there will be several other exclusions as well, such as only inspecting a percentage of socket outlets etc.
The usual problem is that people carry out notifiable works, and then want someone else to submit details to building control. This will NOT happen, because:
1. None of the schemes allow this.
2. If the person doing the work is not a scheme member, they have already broken the law by not notifiying - it should have been notified before work started. Having someone else do the notification later on won't help.
3. There is a lot more to installing circuits properly than just doing a few tests at the end. Other considerations include where cables are installed, what type of cable, position and size of holes in walls/joists, sealing holes in ceilings to block moisture, selecting the correct type of light fitting, installing accessories in the correct places and the correct height, energy efficiency, ventilation, not breaching fire barriers, and plenty more.