What does kill -HUP <pid> do when sent to the nginx master process?

It reloads the configuration without dropping requests. By default SIGHUP ends a process. But nginx, like many daemons (background services), installs its own handler and treats SIGHUP as a reload:
• It reads the config file again.
• It starts new worker processes.
• It lets the old workers finish their current requests, then stops them.

"Nothing, because SIGHUP only affects processes attached to a terminal" is wrong. You can send SIGHUP to any process, whether it has a terminal or not.