You do not sat at what stage you failed but, as a private individual, you can sit ATC licensing examinations if you have completed the requirements in CAP744; you do not have to be employed by NATS.
So why did you fail? Did you not work hard enough at the book work, or was it the practical side which you found difficult? I don't have much sympathy with someone who fails book work but I do sympathise with those with practical problems. ATC is a difficult job which some can do and some can't, irrespective of background, education or the amount of OJT undertaken. I've seen trainees pass exams with 99% but were then too scared to sit at a live ATC position.
It seems to me an awful shame that you may have completed a large part of the course but failed the exams. Presumably you were thoroughly debriefed so you know what went wrong? If you are prepared to talk about this some of us may be able to offer more advice. Having a Degree means zilch in operational ATC. I have just two GCE O levels, neither in maths nor a science subject, but worked alongside people with degrees who were no better than me at doing the job so don't worry too much on that score if your aim is still to be an ATCO.
Non-NATS airfields include some of the smaller places like Kidlington, Shoreham, Isles of Scilly and manufacturers airfields like Hawarden. East Midlands, Tees-Side and some other larger places are non-NATS too I believe, although the situation may have changed. (Someone on here is bound to provide a more comprehensive list).
Good luck whatever happens.