I don't work for NATS, but they provide excellent training and the career structure gives you plenty of options once qualified and the chance to earn some very decent cash.
I don't know much about Eurocontrol but don't forget with NATS you can do Aerodrome/Approach or Area Control, whereas with Eurocontrol you can only do Area Control.
You've clearly done some homework, but I would strongly recommend doing your degree then apply to whichever you think is best for you.
University experience, a degree in anything Aerospace and being that little bit older and more mature will give you a much better chance of coping with the pressures of the courses.
After working your nuts off through Uni the prospect of spending another 18 months or so training may not be everyone's cup of tea and be under no illusion that it is bloody hard work, but well worth it.
With an aerospace degree you can always do something else if you don't like being an ATCO, without it you'll be in a far weaker position.
Best of luck to you.