What happens when an airline pilot decides not to fly at the last minute?
There would be a great deal of variability on what happens depending on the airline, whether or not it was unionized, what country it was in, who the pilot was, and the specific situation of the flight involved.

The scenario of an hour or less is a truly rare occurrence. The system is set up to prevent such happenings. The last two carriers I worked for had a 2-hour showtime. If a pilot hadn't checked in or been picked up by that time, the search immediately started for a replacement.

The only occurrence of an approximate one hour replacement need that I'm personally aware of was when a captain failed a breathalyzer test at his 2-hour show time. Given the confusion of the moment, it was well less than 2 hours when they started the search for a replacement. There was no ready reserve, and no management pilots available that were current, so they had to start stealing captains from flights due to depart later. I was due to operate JFK to AMS around 2 hours after the JFK to TLV flight they needed a captain for, so they stole me off my flight, which meant they had to get someone else for that, etc.

There is good reason for a 2-hour showtime other than just making sure a pilot is there. There is work to be done, and that work has to be done by someone. In this case, a non-current management pilot did that work, including my cockpit flow. I went directly into cockpit, sat down, and called for the before start checklist.

There was in this instance a good reason to do what could be done to avoid taking a delay. Numbers of flights leave JFK for Europe and the Middle East in the evening. If you take a delay, you may well get a lower altitude than planned because someone ahead of you has grabbed it; everybody's going over the North Atlantic track system. If that happens, you need more fuel, which is more cost, etc.

What would happen to a pilot causing a one-hour replacement depends on so many things I won't even try to address that other than to say that it could range from no penalty whatsoever to immediate termination. In the case above, the captain was terminated, and since he was in his late 50s back when the age 60 retirement rule was in effect, it was the end of his airline career insofar as U.S. airlines were concerned.