It would be more convenient for at least some timed quests (for example, the "collect x of y" ones) to wait until you complete the first prescribed action before starting the clock, rather than simply starting it when you accept the quest.
In my case, I got the quest to collect 13 meats from 13 pigs in one game day. I figured I would accept it, so in case I forgot about it, I'd still get credit towards the quest for any pigs I nibbled. The trouble is, I was in the mines at the time, nowhere near any pigs - and then the clock started. So, I decided to go find some pigs and just complete the quest before time ran out.
On a side note, I also do not know if there is any way to find out what happens when a quest clock runs out, other than letting one do so. I figured it probably wouldn't lock me out of that quest forever, but I decided not to take that chance. I've since heard it just resets the clock.
Still, if you lose half your alloted time in transit, you are left with a most uncool choice: scramble and hope you can finish before the clock runs out, or hang around and wait for it to reset. Hence, the game should wait until you are actively engaged in a quest (whatever that may mean for a given quest) before starting the clock.