So, summary of what ended up happening.
Seems that Connington was indeed closed, and that means you can't land there (except in emergency etc). Sywell (actually, Northampton Sywell, nice long asphalt runway) was open, but operating as unlicensed. This would have been fine to land at, but the CFI said there would be much less training benefit from it, so cancelled today's lesson.
Apparently, blind calls on the published frequency, treating it as an uncontrolled field is the way to go.