Once an early access game has been purchased on Steam, how do you find it in your library to install it? Do you have to enable "Beta participation" in Steam? Or do you have to wait a couple of days before the game shows up in your "Game Library|All Games"?
You don't have to enable anything. It shows up in your library like any other game you bought.
You simply install it by double clicking the entry. If it does not show up after a successful purchase, try to restart Steam and/or try to rename ClienRegistry.blob to force a redownload of your licenses.
If the problem persists, contact Steam support.