Only one time have I been diverted - LAX-DEN, diverted to COS due to low ceiling at DEN. We landed at like 130am.
Now DEN-COS is only like a 40 minute drive. But there's a myriad of issues with this, first being it's 130am and the airport is closed. The (compass? I think? this was AA express, not UA) called their office and tried to figure what to do - it sounded like they might have to actually fly back to LAX (I was on a MR, so figured free segments + miles, not complaining at all).
They eventually got a hold of the airport manager and got a shuttle bus to bus pax up to DEN since there was about to be an uprising on board. I just got rebooked COS-DFW-DEN.
To answer your question though, a lot of it depends on the situation, and there isn't going to be one easy answer. The carrier is responsible for getting you to your destination, unless there is a mutual agreement (i.e getting off in MKE and changing your rental car plans).
Depending on weather and other things, the plane usually needs to be somewhere, so the plane won't sit in one place too long unless it's a big mechanical issue that caused the delay (in which case there is usually a ferry flight flown in). Depending on how far away the diversion is, shuttle buses are pretty common. Otherwise the plane usually refuels and takes off eventually.