HE hides the execution complexity from the system doing the computation, so no, it can't do someone else's computation and just wait in between to avoid leaking

HE hides the execution complexity from the system doing the computation, so no, it can't do someone else's computation and just wait in between to avoid leaking information, it's designed so that the computing operation order and quantity is simply independent on the input data, i.e. the worst case complexity, and a valid HE scheme would have mathematical proof that it's impossible for the system to find a way to do it faster than the worst case.