Speaking from a 20 year military controlling background, if it's pure controlling aircraft you want I'd definitely recommend you apply to Eurocontrol or NATS first. Not because the military is bad (far from it) it's just most military types who wish to climb the career ladder only really control for 2 or 3 postings (as an officer). Then it's off to some admin/policy/bag carrying job in order to 'progress'. Horses for courses though, depends if that floats your boat.
Also the military 'license' isn't a license at all, after a career in the military you'll be an experienced yet unqualified controller.
In the civilian world you are ostensibly trained by the company then examined by the CAA for your licence, pay is slightly better too.
Competition for places at NATS and Eurocontrol is extremely fierce. Where as with the military there is still a threshold to reach in terms of aptitude tests but we are desperate to recruit. So if you have trainable controlling aptitude and trainable leadership skills you are in!
Don't worry too much about the military training part, if you have anything about you, a good attitude and level of fitness the instructors will get you through.
Good luck