More often than not, the requirement for a degree is to satisfy visa conditions. Unfortunately, if you don't satisfy these conditions, you won't be issued with a work visa and thus cannot work legally in that particular country. That said, once you have five years' relevant experience, some countries such as Vietnam, will issue you with a work visa even if you don't have a degree.
The list of countries where you can legally work without a degree is rapidly decreasing. Currently, your best bet is China, and I suspect that you could still find (legal) work in much of South America without having a degree. I know Ecuador doesn't require a degree. Of course if you are in possession of a British passport (or another EU member state passport), the whole of the EU is open to you. Some schools will still insist on you having a degree, but there are plenty that won't.