There is a discrepancy between the Highway Code and the Law here !
The Highway Code could clarify whether it forbids cycles overtaking or being overtaken on crossings, by using the distinction between 'vehicle' and 'motor vehicle' precisely.
Actually, I would play safe, and leave the Highway Code as-is, for safety's sake.
It will be safer if cyclists neither overtake, nor are overtaken.
But we should definitely educate Driving Instructors, and also those who train and assess Driving Instructors, to err on the side of safety, instead of nit-picking about what are probably unintended loopholes in the Law !
Pedestrian crossings overtaking a cyclist ? - [ adiforum.co.uk ]
Contrast:
Highway Code 191
You MUST NOT park on a crossing or in the area covered by the zig-zag lines. You MUST NOT overtake the moving vehicle nearest the crossing or the vehicle nearest the crossing which has stopped to give way to pedestrians.





