Mahogany (on Turl Street, Market St & Little Clarendon St) are supposed to be good. They're quite pricey I understand, but they do a cheap day for students I believe. I might be wrong though...I just know several friends who went 2years ago and said although it was pricey it was good.
Also depends on what your definition of cheap is - ie. whereabouts you're from in UK! At home with me it costs £30 to get a cut at a fairly average salon!