Your present Profession / Branch would be useful for a more complete answer, but I'm guessing by your opening statement of a complete career change it isn't ATC.
Is it possible to qualify as an ATCO part-time, at my own expense and whilst in full-time employment, or is the only viable solution to find a sponsor, leave full-time employment and train full-time for minimum/no salary?
The self study route closed in 1994 with the introduction of RGAT. Your only option now is a full time course or courses. At your age, and in the current climate, it is doubtful whether anybody would sponsor you, especially if you have no ATC background in the Military. (A lot of those guys are self sponsoring, as I did 16 years ago).
What are the career prospects of a minimally qualified but inexperienced ATCO in the private sector?
I hate to say this but, without any previous experience, and with age not on your side, not fantastic. You would have to be capable of a very hard and convincing sell. Without wishing to appear too brutal, you might want to be looking at things more closely aligned to your current skill sets.