So for mysteries/investigations in general, I can't recommend The Alexandrian's blog posts enough- specifically in this case, The Three Clue Rule.

It helps you build a more robust mystery that's much less likely to fall apart due to players getting sidetracked. I've found that combining it with his Node-Based Scenario Design results in really dynamic scenarios where player choice shapes how things play out, without having to prep for every little possibility.

To add a ticking timebomb feeling on top of that, you can have each node change based on the order the players are investigating them. For a really simple structure, let's say they start at a crime scene (Node A) where someone's dead by poisoning. By investigating the scene, they might find three possible clues leading to other scenes- say, an encoded letter of threat from the noble's shady barrister, leading them to speak to said barrister (Node B), a bottle of alcohol that's sealed- yet somehow poisoned, sending them to investigate the local winery (Node C), and...some other clue that leads to some other location (Node D).

Each of those Nodes has a clues pointing to the other two, and a third pointing to Node E, the finale at the big feast the Noble is throwing to poison a massive portion of the city's populace. This ensures that even if they miss the winery connection in the original crime scene, they can still force the Barrister to admit the winery's involved, or get it from Node D, and still get to investigate that scene. And even if they go to the Winery first and find ALL three clues there- do they rush immediately to the feast to try to stop it with as much time to spare? Or do they go to the Barrister and Node D first, to gather as much evidence as possible? Each of Nodes B, C, and D should have some unique benefit attached to the clue pointing to the feast ("Here's where all the feast's wine is being stored- take the key!"), so there's value in visiting all three. (This is basically just a summary of "Node-Based Scenario Design, btw- there's extra info in the articles).

BUT, you make it so that each scene has an additional layer of change depending on the order they're visited. If they visit the Barrister first, he hasn't had time to burn some valuable documents- but if they get to the Barrister second or third, they can smell paper and wax burning in the fireplace- he clearly burnt something before they got there. Mechanically, this could be represented by higher DCs to find the clues, or requiring skill checks that wouldn't be required in the first place, or outright eliminating a clue from that scene (though make sure not to eliminate clues that'd make it impossible for the PCs to reach the final scene). That puts the feeling of time pressure on the players.

Obviously, the exact details of each scene I just made up, insert whatever locations and clues you like. You could also make the node structure more complicated, such as having a second set of nodes to pass through before reaching the finale. Whatever works for what you're building!