Does singing a Billy Bragg song require putting on a totally naff, over the top pseudo- East End accent, an accent which disappears as soon as BB sings Woody Guthrie or just speaks?

Sing in an accent/dialect/pronunciation which you feel comfortable with. If you're not comfortable with these aspects when you're singing - and I'm not talking about content- then it's very likely that your audience will notice this, to say nothing of a performance which doesn't do the song justice.