Don't think I have ever seen a lighting circuit protected by 20A

Not unknown - I've seen a few 1980s office blocks with 20A lighting circuits with 20A grid switches and fluorescents with fused terminal blocks - usually in 2.5mm² though.

I was told by this highly suspect electrical company that as it's a industrial site, it doesn't need to be signed off.  Not sure that is correct.

Partly - all the troublesome Part P of the building regulations and the corresponding registered competent persons scheme and notification stuff is only for domestics. For everything else it's vanilla BS 7671 plus however the local duty holder decides how to demonstrate how they're meeting the Health & Safety at Work legislation and the Electricity at Work regulations (and associated legislation, and possibly insurance company requirements too). A sensible minimum is likely to be BS 7671 certificates signed by whichever competent person did the work. A lot of companies like NICEIC or similar registration as a means to demonstrate that those doing the work are competent (or if it turns out they're not, there's someone else to blame!)

   - Andy.

Assuming you are happy to do the testing, mainly verify insulation, earthing etc and sign on the minor works cert that is is OK, that in legally (maybe actually more than) enough. No need for anyone to demand to see your certificates or to pay club memberships.
But the buildings insurance or the local management may care, and impose house rules, anti-discrimination rules do not allow you to be barred on your age but an arbitrary 'NICIEC only'  requirement may be in place. You will have to ask.

It does not have to be your day  job... As an example I neither belong to a part P scheme nor the IET, but I have designed and installed a fair amount(!). I just have to jump hoops for domestic local authority notifiable, so tend to avoid that except for family and friends. Equally, (as I have the required knowledge and the self confidence, and access to the test kit), I have done plenty of new circuits and things for scouts HQs and similar situations (as a charitable organization I strongly support) where there is a local ownership of the buildings and some common sense can be applied, and as required I then fill out my own certs for the duty holder and exec committee to refer to. For me the day job tends to involve larger project teams, and I may draw things in the plans, but someone else can do it then.
Mike.

Assuming you are happy to do the testing, mainly verify insulation, earthing etc and sign on the minor works cert that is is OK, that in legally (maybe actually more than) enough. No need for anyone to demand to see your certificates or to pay club memberships.
But the buildings insurance or the local management may care, and impose house rules, anti-discrimination rules do not allow you to be barred on your age but an arbitrary 'NICIEC only'  requirement may be in place. You will have to ask.

It does not have to be your day  job... As an example I neither belong to a part P scheme nor the IET, but I have designed and installed a fair amount(!). I just have to jump hoops for domestic local authority notifiable, so tend to avoid that except for family and friends. Equally, (as I have the required knowledge and the self confidence, and access to the test kit), I have done plenty of new circuits and things for scouts HQs and similar situations (as a charitable organization I strongly support) where there is a local ownership of the buildings and some common sense can be applied, and as required I then fill out my own certs for the duty holder and exec committee to refer to. For me the day job tends to involve larger project teams, and I may draw things in the plans, but someone else can do it then.
Mike.