When the Radio Antenna Tower light goes out...

What happens? How long does it take to replace it? If the tower is in the middle of nowhere, does it take a long time?

The FCC has rules concerning antenna tower aircraft warning light maintenance.

You notify the FAA, which then notifies the FCC. Operators have 15 days to fix it. If not fixed, they must re-notify the FAA. A Cite

Some station engineers will tower climb. Some won’t. There are contractors who specialise in tower building and maintenance. They are a different breed of person.

Operators are required to check the lights periodically. If not visually, then its done by remote sensing.