"an airfield that has ATC coverage at weekends, say Duxford or Cambridge?"
Cambridge has ATC (i.e. Air Traffic Control with Tower, Approach and Radar call-signs). Duxford, like most small airfields does not. Duxford is an Information call-sign so the AFISO controls ground movements. As soon as the aeroplane is on the runway, it's uncontrolled and up to the pilot. Mind you, even the people that run airfields don't always know the difference, calling Church Fenton by telephone this week to see whether they operate after dark, one of the telephone options is for Air Traffic Control when the airfield has at best an air-ground radio station i.e. not control at all.