As for a sense of urgency. I don't particularly care either way. A sense of urgency is great for creating tension and an atmosphere, but that is also at odds with exploration-based gameplay. D:OS is a game about freeform exploration, and putting urgent timers on everything would kill that part off. The boat with the Divine Magisters will arrive "tomorrow", but tomorrow will never come. It doesn't fit with the desired gameplay experience. Freedom and urgency are opposing forces in terms of gameplay.
You can only put a sense of urgency into D:OS by restricting or removing some of the freedom you get. The Luculla Mines are the key example. You lose the ability to travel or escape in a pretty arbitrary manner basically for the benefit of a two-minute or four-turn escape sequence. It was a little clunky.
I think the best way I've seen urgency used in a game is by giving a choice. You can either do 1) Save this town, or 2) Go slay this monster. You can't do both. By doing this you are forced to make a meaningful choice, that changes the game going forward, but you can take your time in making that choice. So you don't have to be in a hurry to make the choice, but when you do someone lives and someone dies. I also like how this adds a lot to replay value.
Though I haven't played it myself (it's on my wishlist) I think Wasteland 2 had some choices like this.