Many universities and colleges offer Game Design curriculum. The most famous is probably DigiPen, which is located in Redmond, Washington, literally on the same campus as Nintendo of America. I think they're mostly an advanced learning school, though, so you'd need a bachelors in one of several degree paths before they admit you.

https://www.digipen.edu/

Many art colleges have very detailed tracks created around the design and content aspects, but are not very reliable when it comes to game programming. If you're interesting mostly in the design of game concepts, programming, etc., you'll want to go to a college that focuses more on the engineering.

Carnegie Mellon has two schools that have tracks.

http://www.etc.cmu.edu/
http://www.hcii.cs.cmu.edu/

Other engineering focused colleges with Game Design schools:

http://gdiac.cis.cornell.edu/
http://www.gsas.rpi.edu/
http://www.cs.du.edu/curricula_general

Most of the curriculum is design around IGDA's recommendations, who also hosts national contests and events.

http://www.igda.org/

Gamasutra, one of the most popular hubs for the game industry, also has an education site that has an extensive list of schools, but keep in mind many are community colleges or trade schools. I doubt you want to make the effort crossing the pond, getting a visa, etc. to not go to a well known university.

http://www.gamecareerguide.com/
http://www.gamasutra.com/

Another good way of learning about what schools are well respected for programming, art and design in the gaming industry is by going to a game company's website and browse down the bios. For instance, Firaxis, here in Maryland and home of the famous Sid Meier, have bios for their employees. If you have a favorite design house, check their website for the same.

http://www.firaxis.com/company/bios_home.php

As far as what you need to get into the country, the admission offices for most schools will be very helpful explaining the process. I'd contact the top two schools you want to goto and ask them both for help.

Having a portfolio really helps as well, so if you've done any playing around with game creation, even if it's tiny things like ideas, sketches, etc., it wouldn't hurt to get them together in some snappy way to impress.

Also, keep in mind that any decent game designer knows how to program at least a tiny bit, so expect to learn it and don't take it lightly. You'd be expected in the real-world to build tiny demos of your game concepts and you'll probably be on your own doing it. Getting use to doing this in college is a big bonus.