I'm not aiming to be negative here, but here it goes...
Games Designer isn't the job people think it is, so get that out your head. It really requires indepth general knowledge about all things game creation related, usually involving a strong skillset from every department possible to assist in development, which is incredibly technical. But the people who actually make the choices on a game are the design leads, who worked as game designers for years.
Games testing falls to the passionate people, but really most of the people who do it are younger software/game students. Game testing isn't a full time, supportable job... and most of the time your running around in say one room looking for every single possible glitch/bug you can find. Technical understandings of the bugs are usually a given. Its not fun by any means, and you play games before they look anything remotely like a game. Many game-testing opportunities fall to local colleges/universities as a sign-up and earn £20 type of thing.
Online training sites will be a pipe-dream I'm afraid. Even most university courses.
To get to America, someone will need to vouch for you. You can't just waltz into the country and start job hunting. You either go there on an agreed working visa at a job already assigned, or you get access to a green card.
If you are proficient on a computer at either a programming level, artistic level, technical level and develop a portfolio of things you have learned and what you like doing, then degrees and education mean nothing. They will take passionate people, who show commitment. A degree will just push you in the right direction.
Start with learning to program, and see how well you can pick it up! Java would be a fine starting place.
However, realistically your best bet would be to find a small and independent studio near by (should be a decent handful in London) and walk up all determined with your CV and a well written cover letter explaining your situation. Ask if they have any positions available that you might be able to work on, even as solid work experience and start from the ground up. Just make sure your carpentry business is still generating you some money!
EDIT:
On the other hand, please take note that there is more qualified students walking out of university looking for jobs in the industry than there are positions to take... so you'd need to show your passion.
Another option would be to look for someone who can code games, someone who does art, and then try rile them up to develop a game, where you get to do the idea-factory role and get them to teach you to do little odd tasks. You can easily make a small mobile game like that or even a small PC/XBLA game. Finding the partners is the tricky job here...
At the end of the day, I'd trade a huge portion of my knowledge for top notch woodworking skills, as I would love to make electric guitars :D