There is year-round, decently-paid TEFL work at some British universities. Some international students have to complete a foundation course to get their academic qualifications up to speed before they can study on an undergraduate degree, and there are other pre-uni course options and requirements, for both UG and PG students, all of which include an English language component. Therefore, TESOL teachers, of an EAP variety, are needed to teach them. Although it can be hard to get into it and secure a permanent position, it isn't impossible; it can be really rewarding if you like that kind of thing, and I'd say it's a lot less stressful and fraught with bureaucracy than [s]proper teaching[/s] teaching in the state sector.
Most universities require a Bachelors degree, an initial teaching qualification, e.g. the CELTA, plus a Diploma and / or a Masters in TESOL, plus experience, probably 2-3 years' minimum, so you're nearly there - take Alex's advice and spend another year or so teaching abroad (it's more fun). Pre-sessional programmes are a good way to cut your teeth on the EAP scene, and they might be a bit more flexible in terms of what qualifications and experience they ask for, because the demand for teachers can be quite high (FYI they're in the summer, just before the university year starts).