There as obviously been a big breakdown between you and the electrician for whatever reason, be it the fact that he left work to be done by people you have considered to be non-competant persons. As you are paying the bills you are quite within your rights to object to this, and ask for an explaination and proof of thier competancy.
Of course if your not happy with that explaination, you have every rights to stop the work and ask the electrician to leave the site. This though IMO should be the last resort as it opens up all types of litigation and expensive procedures. I think it would have been more advantageous to sit down and talk things through, but it seems that was not done by either of you.
That though seems to be the case, you have "paid" them off and now you want the work completed by another contractor. That in itself is not going to be too difficult to do, but it could be expensive. There are many times, espicially in todays market, when a job is started by a contractor, who goes bust, and the work is finished by another.
Whoever you call in will advise you, or should advise you to contact your LABC. You will need to explain what happened, that you originally used a NICEIC contractor who was going to notify his work after completion, but due to circumstances, which you explian, you felt the need to stop the work and ejected them from your home. You advise them also that you have contacted another scheme affliated contractor who is willing to take on the work, but obviously can not sign off the 1st fix work, but is willing to do a PIR on that part of the installation and a full EIC on his part. The chances are the LABC will accept this, and you will get your work completed. The new electrician may also do a full EIC for all the installation, but just part 3 of the EIC will be completed by him for the Inspection and Testing, it can only be sorted out through the LABC, you and the new contractor.
Cost? Well that is the crux. The new contractor is going to thoroughly check the existing work, he may even decide that it is sub standard and remove it, unfortuantely this is the penalty you face for ejecting the original contractor, this is why I originally said that would have been my last resort. You can of course claim the extra expense of the removal of the existing installation through the courts, and he will counter claim that you never gave him a chance to rectify the work. In all honesty I think it will cost you more in the long run in expenses and stress, as it could drag on and on. It may very well mean you bite the bullet take the loss and move on.