provide staff for the contract Towers (mainly small VFR airports)
Contract towers are not necessarily "small VFR airports". Oshkosh, for starters, is a contract tower, with well-used instrument approaches on the 4 main runways. I think the company is Midwest, but when I did a google search, it came up with Midwest ATC and I'm not sure thats them. I'll check around for you and post back.
Have you tried searching on the NATS and FAA websites to see specifically what they require with regards to green card vs citizenship? I know there is a British person working in Toronto centre, but haven't heard a British voice on the US ATC system, yet. Might it be easier to move to Canada, and work for their ATC?
Of course, you will have to be fully trained at sleeping on the job if you move to the US
