Ask for a fuller explanation of their decision.
What has belt tensioning got to do with Health and Safety?
If the belt is safely contained within a housing, a break in the belt due to over-tensioning will not threaten anyone in the vicinity.
If the belt is under-tensioned, it will slip, making it difficult to turn consistently.
If any belt adjustment is possible with the power on, a simple electrical interlock would solve that.
Quite literally millions of ventilation fans are belt driven, with a belt adjustment mechanism, but no spring loaded jockey wheel or similar - considered perfectly safe.
Rubber belts have an inherent elasticity which gives a small range of automatic tensioning i.e. not "hard" and if the lathe is designed for the belt to be tensioned by the weight of the motor it is working correctly.
Finally, what regulation is being cited for this HSE decision?
Super-finally (!), what experience does the HSE inspector have in this area of expertise?
No wonder schools are giving up on trying to teach any practical skills.
Duncan