From the product page, select Related items, then select My Add-Ons. On the PS4, go to the PS4 menu, and then navigate to your Library. Select Manage game, and from the Ready to Install column, select the Automatron add-on. On the Xbox One, go to the Home screen, hold and press to get to the My Games and Apps menu.
You might need to do this for both the Automatron DLC and the Season Pass. The download should start on its own, but in case it doesn’t, you might want to uncheck and re-check the checkbox again. Right-click on Fallout 4, select View Downloadable Content and make sure that the ‘Install’ checkbox is checked for the Automatron DLC. If you have Fallout 4 on Steam, then first log into your Steam account, and go to your library.