I always read the flight delay updates on our intranet system and so often the reason given is "Missed ATC Slot". Obviously we've taken too long boarding or fueling or the crew arrived late or whatever. So what happens on your end?
Simple answer - not a lot. If a flight misses a slot then they have to delay their FPL to a realistic time of departure, they also have to accept the next slot they're given. This might be brought forward by submitting a READY message but that's not always the case. We, as ATC, have a 15 minute slot "window" -5/+10 minutes we can use to facilitate the traffic sequence, occasionally we might be able to get you an extra 5 minutes on top of that. But if the flight isn't going to make that, then it's up to the crew / Flight Ops department to sort that out.
Local ATC have no control over the slots issued - that's all done by CFMU. As such we're purely the middle men involved.