I have to say I'm surprised that the instructor did not require each student to have their own timing device. That's not very good instruction; it's important to learn to time your own dives. I use a dive watch with a bezel on every dive, even ones that are shallow enough to not warrant a computer.
But, to try to answer your question, lets say you're on a dive and your computer is your only timing device, and it goes belly-up during the dive. In this case, you time your safety stop by using your buddy's timing device, as long as your buddy isn't olyuser, in which case apparently you'll have to 'compensate him for the liability'.
If by some chance your buddy is missing or a jerk that won't give you the time of day, you could as a last resort start counting. A nice slow count to 200 ought to do it.
Edit: Oh yeah, you don't need a compass on every dive. Only dives where there is some navigation involved that requires one.