According to Truth in Science, Healey was formerly a Fellow of Selwyn College and Lecturer in the Engineering Department at Cambridge University. It adds that "Dr Healey now teaches mathematics at a leading (unnamed) independent school". The school is understood to be Loughborough Grammar School - see http://www.loughgs.leics.sch.uk/docs/Staff.pdf.
This is an independent school (i.e. not part of the state system).