Salutations,
I have found the key with such situations is to start slow and then start racheting up the timing of the events.
I have also found out that splitting up the party is essential- and ending each players turn on a cliffhanger.
Ok, example time:
Walled city in the middle of a torrential rain storm- and the city is flooding.
Word reachs the adventures the main wooden gates to the city are closed- this will cause the water to raise too high. The pc's need to get there and quickly.
On their way- they see a small child swept into a sewer opening. There goes one pc to help. Along with dealing with whatever monsters are desperate in the flooding sewers- the rats are going nuts, the tide is strong, and there is a kid to save.
Further on- they spot a noble tower is on fire. A young woman is screaming from help on the top floor. There goes another pc. Not only is the courtyard of the estate already underwater, but the tower is an inferno.
A bit further- some villians have taken the chaos as an opportunity to rob a bank/guild/whatever- there goes another pc or two.
Finally- the gate.
How do you get the party to split up? You meta-game a bit and tell them they realize they can not all stop to get involved in whatever incident, so they either split-up, ignore it, or move on.
You can also personalize the events. For example: The kid was a local beggar who had previously attached themself to a pc. The noble-girl is the object of flirtation of another pc. etc. etc.
At first, you should be spending a few minutes per pc. As things get more intense- you spend less and less time with each pc. The climax of every little storyline should end at exactly the same moment.
That is the "dealing with the disaster"-like scenario. There are other scenarios- escape, combat during, or possibly even trying to stop the disaster.
All of them share the same timing issue- you need to make things intense and rushed. Don't allow time to consider all the options.
If your players would be up for it, then you could tell the players before the adventure that at time decisions will be limited to # second time limit. If they don't decide in that time, then they don't do anything in that round. Time is of the issue- describe how events get worse as they sit around.
Good luck!
FD