The Safeguarding Vulnerable Groups Act 2006 applies to anyone who works in a regulated activity with children or vulnerable adults on a frequent or intensive basis (my emphasis). Whilst it is true that flight instruction is a regulated activity, it is only subject to the Act if frequent or intensive contact with children or vulnerable adults is involved.
For a flying instructor or his/her employer to be prosecuted under the Act, it will be necessary to show that he or she has individually worked with a child or vulnerable adult on a frequent or intensive basis without complying with the requirement for a CRB check.
As the TrainingCom makes clear, the CAA has no involvement in the administration of or verification of compliance with the provisions of the Act, nor is there any reason for it to do so.
This will doubtless affect some instructors (e.g. those involved in Air Cadet activities, flying scholarships, etc.) but there is no reason for a general panic. As BEagle suggests, the simple answer is to ensure that all of your students are adults.