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In fairness it's your choice to have only NICEIC but it's a vast oversimplification of the regulatory requirements. Any notifiable work can be signed off by a representative of the local council or registered 3rd party (prior arrangement before the work commences is necessary) or by any registered electrician, they could be registered with NICEIC, NAPIT or Stroma or there are some smaller registration bodies.
Electrical Installation Condition Reports also require an electrician to have a certain level of qualification and experience, usually C&G 2394/2395 which is contained in the full electrical apprenticeship (or C&G 2391 the older qualification) plus you need a specific insurance type and registration with a professional body that recognises your skill in carrying out EICR's. Part P has little or nothing to do with it and is a far easier qualification to obtain.
However most electrical work is not notifiable, CU replacement, adding a new circuit to a CU some garden and some bathroom electrics are. I'm still simplifying it but there's not always the requirement for notification.
It's true that anyone working professionally on your electrical system should have a good working knowledge of BS7671 and Part P and follow those rules and regulations, in which case they should probably have the up to date qualifications in BS7671 and Part P.
Get a Part P qualified sparks in to do an EICR (electrical inspection condition report) and any remedial work (if necessary) to make it safe. No one but the qualified original installer or the council's Building Regs person can sign off the original work. That EICR and proof of remedial work is the next best thing to the original.
We had the same problem, did the above, and sold our flat without any problems. Oh, and I threatened to take the original contractor to the small claims court - which I was fully prepared to do - and after six months of silence he was soon round with £400 in an envelope and a bucketful of excuses. From now on: no Part P qual/NICEIC number, no work.