For all that schools aren't usually involved, the Doctor has a... not inconsiderable amount of experience in wandering into places he really shouldn't be and coming out at least relatively unscathed. Perhaps not unnoticed but he can usually manage unharmed. Admittedly, most of those don't also involve a bomb that's going to be going off at some point in the next hour, but he's nothing if not adaptable.

And he absolutely means to do everything in his power to make sure that the bomb doesn't end up going off.

"I'm going to assume that someone would have run into what we're looking for already if it were out in the open, so we can likely rule out any of the commonly traversed areas. Or at least, the parts of same that someone would be looking at."

Though they might need to check the undersides of desks also.