If you just want to get your foot in the door as an EFL instructor you can do that with any degree. The gold standard in TEFL is the CELTA certificate which is offered at centers around the world but, just to get your foot in the door, any old rinky-dink TEFL certificate will do.
That said, if you want to work in good international schools, you're going to need a real teaching certification. This is where Britbongs are in luck. While there are online programs to help American and Canadian teachers get certified in their home countries, there seem to be a lot more online and distance PGCE programs targeted at teachers currently in Asia, and from what I can tell, they're just as respected as a qualification done in your home country.
If you still need a degree, why not look at doing it in Asia? Nottingham University has campuses in China and Malaysia. You could probably even make some money tutoring on the side even if it's technically not "allowed" on a student visa. Likewise, if you're already making good money while working remotely, why not just relocate to Asia while getting a degree through The Open University or some other British uni offering online courses?
