Am I right in thinking that a degree is absolutely essential to get a work permit for teaching English in Thailand? Or is it simply advisable and dependent on the office one applies to?

I have 'A' levels (high school education), plus a college education- HNC Electronics, a TEFL certificate, over a years teaching experience (from Thailand - cough, ah hmm).

I know I can get support from a school in BKK, and so enter Thailand with a non Imm O visa, but am I likely to get the w/p from the BKK office?

No, it's not absolutely essential, although you may find it is

While it is possible for schools to get you a WP without a degree (or comparable) at lot of them can't be bothered to find out how (certainly here in Bangkok anyway). If you go to the teachers room part of the board here and go a few pages back you'll see a 7 page long thread where Stickman, Sunbelt (or was it Indo Siam) and myself all say the same, it is possible, but not easy!