The reading for this is pretty weird. 計 is all nice and normal, but 時 is と for some reason. It probably is coming from the vocab word 時 (とき), is my guess. But even if it is, it's shortened, making it a step more complicated. Maybe it's just better to imagine that your watch is a walky talky (とけい) and run with that?
Koichi dropped the clock Viet had given him from the ship.
We're gonna meet at the clock tower at 3 p.m., okay?
Hey, isn’t that spit on your watch?