Once all files are downloaded, the game will notify you with a popup. You can track the download progress in-game in the menu Options > Audio Language.
You can already start playing in English with the subtitles you selected (or without, if you did not select any) while the download is happening in the background. The game will then start to download the necessary files. If you select another language than English, you'll be briefly redirected to your console's store to start the download. You will be able to choose from all the languages available in your version of the game. After first launch and download of an audio pack, you can switch languages by opening the Options > Audio Language menu. On first launch of the game you will be prompted to decide which audio language you want to play the game in. To do that, make sure your platform is connected to the internet (a high speed internet connection is recommended each language pack is about 450MB (PC) / 660MB (console) in size) and follow the process outlined here: You simply need to download the pack for your desired language first. If you wish to play the game with another audio language, you can.
The menu and subtitles will be available several languages included in your game version. Whether you own a digital or physical version of Assassin's Creed: Origins, it will contain only one audio language at first: English.