At least on small aircraft, the HSI is slaved to the NAV1 radio. The NAV1 radial is selected with a small knob of the HSI. Now, when I want to fly an ILS approach I don't need to set a radial myself as an ILS / LOC has only one. But if the runway has a different orientation than the radial / CDI needle o the HSI, this is confusing. Do I need to set the correct radial myself, or does the NAV radio / autopilot do that automatically ?
It depends on your instrument suite.
If you have just a radio with connected HSI, you have to set the course yourself, because on the radio you only set the frequency and there are hundreds ILS installations with the same frequency across the world.
If you have flight management system with a waypoint database where you set the ILS by its identifier, it will set the course for you. Airliners had that for some time, but integrated avionics suites are making their way to small airplanes as well that have it (together with full RNAV capability where you just select waypoints and the HSI gets slaved to the GPS).
Yes, you need to set the correct course yourself. There is no servo motor inside a mechanical HSI that can receive a command from an autopilot to change the course for you, it is designed to be manually set by the pilot.
In my experience in GA with KAP140 autopilots, these are very basic non-integrated "add-ons" that need a lot of careful monitoring as they blindly follow the course needle (whether you set it up correctly or not). These basic autopilots are not sending any data TO any flight instruments (including the HSI). As you know, when you engage NAV hold the autopilot simply turns the airplane to intercept and center the needle. It's pretty limited in its capabilities.