I'm not being rude, but be aware that your Mil' experience counts for little when you are doing the civvi' courses, it's best to cleanse your head of Mil procedures and accept what you are being taught wherever you go.
To answer your questions:
1. If you are accepted by NATS, they pay for your training and pay you a minimal salary whilst you are at the college. With your background, common sense may prevail and they might stream you into the specialisation you prefer and not what the personnel bods/number crunchers say is required at the time.
At Cwmbran or ASTAC, you choose which courses you want, (and I would recommend APP as well as ADI/APS as it's still a requirement at many non NATS airports - although this may change), but you pay for everything, right down to the pencils and paper.
The cost for ADI/APS is around £30K and you don't get paid a bean, the additional, and biggest risk with this route is that if you fail the course you don't get your money back.
2. Rather an odd question - a licence is a licence and the CAA courses are all taught to the same standards wherever you go to train, so to answer your question, no, non NATS airports don't give a stuff where you are trained.
Your competition for a place at non NATS airports will be fairly substantial, (from their own staff who they've sent on courses to become ATCOs).
In your shoes, I would approach a few airports where I'd like to work, tell them I am funding the courses and ask them for a job when qualified on the basis that they fund some or all of the costs.
Make sure you've thoroughly researched the airport first, the future at places like HUY isn't exactly rosey and remember (I only say this because a small minority of ex Mill ATCOs find it a shock) that in the civvi world you start at the bottom of a big heap and no one will call you Sir.