Long delays are ordinary rather than rare. If you have a connection, the useful window is the first twenty minutes, before several hundred other people reach the same conclusion you have.
The order is: work out whether the connection is genuinely gone, get in a queue and on the phone at the same time, and know what you are asking for before anyone speaks to you.
The first twenty minutes
Before anything else
Check the actual arrival time against your connection time, not the delay figure. A ninety minute delay on a three hour layover is not a problem.
If it is tight
Join the transfer or service desk queue immediately. You can always leave a queue.
While you stand in it
Call the airline. The phone queue and the physical queue run in parallel, and whichever answers first wins.
At the same time
Check the airline app. Rebooking is sometimes available there before an agent offers it.
Before you reach the desk
Decide what you want: the next flight, a different routing, or an overnight with a hotel. Vague requests get vague answers.
Say what happened, once, in order
Whoever you reach can only help with the actual sequence of events. Give it once, cleanly, then say what you need. Leading with the ask before the facts almost always makes the conversation longer.
Keep it to two sentences. Your inbound flight is delayed, you will miss a connection at a named time, and you would like to be rebooked.
Before you walk away, get two things
The name of who you spoke to, and a reference for the conversation. Not to complain later, though it helps with that. The real reason is that if this is not resolved by whoever comes after them, you can pick it up where you left off instead of starting again.
This is the single most useful habit in travel disruption and it takes ten seconds.
What about compensation
A great many people never claim for a disruption, usually because nobody told them they could. It is worth looking into afterwards.
It is deliberately not part of this guide, because what you are owed depends on where you flew from, which carrier, and sometimes which fare, and a confident wrong answer at a desk puts you in a worse position than no answer. Sort out the travel first. Look up entitlement later, when you are sitting down and not at a desk.
What to have ready
4 things to do
Ticks are for this visit only. Nothing here is saved anywhere.