I am not sure about the size of your gym and the way it is organized. Since you are describing this person as ‘the other staff member’, it sounds to me that you are part of a hierarchy, and there should be a person in charge who oversees the staff and who should have a process n place for such eventuality.
If it is club policy that there always has to be a person on the exercise floor exclusively to assist members in general, then I would recommend that those three clients being called with a cancellation of the assessment along with an explanation, an apology and suggestions for rescheduling. It can be explained as a matter of safety for all club members.
If you are a small club, then the owner should establish a standard practice for cases like this so that you won’t have to wonder the next time around.
If you show up to work as a personal trainer, then deal with your clients first. Isn’t there a fitness director/owner who can pickup the slack?