boffo, the student log the flights as solo, i.e. P1, the instructor logs nothing except the solo authorisation (if FI(R)), and that's not needed when the instructor is unrestricted, he/she just lounges around, or better still takes another student for a lesson.
Send Clowns, the CAA form for the removal of FI restrictions (Form FCL676 Issue 2 (SRG\1133)) has a sheet (page 3) which is a record of your supervised solo authorisations, each flight is signed off by the supervising FI, usually the CFI.