There was a star trek enterprise game that was delivered with a cassette.
You put the cassette into a player and started playing. Everybody had his mission, so you were running around in the ship, trying to get it working again.
Every now and then, the cassette would say things like "Hey you, there, who's turn it is: Report in to the bridge!"
. Then you had to do what the cassette told you, before you could go on with the real mission.
So, everybody was keen on playing as fast as possible and handing the dice to the next player to avoid this.
I don't see this in HQ, somehow. But it would be fun to try once.
Time limits can make every oh-so-simple mission complicated. Sure you can count the fields between entry and target room, divide by seven which would be the average number to roll, then add one turn for every small fight and two turns for every big fight you plan, and give it that as time limit to arrive somewhere.
But those time limits they can also kill the exploratory part of HQ when the players press on too much, and it could change the general feeling of the game. One of the critics on our only DESCENT successor is that it is a race all the time.
I would say, this is a nice thing to do once in a while, as you said, with some storyline behind: The
cultist finish their evil ritual in the
sacrifice chamber... No i am not working for Thantos i just like his models.
My personal time pressure comes from my Evil Wizard Deck. I adjusted the rate of cards drawn to just punish excess slowness. It's sufficient. They don't do slow turns anymore.