Is there any way to know if the update downloaded? I was not prompted to download it, so is there a possibility that it has already downloaded automatically?
You would not be able to start and play the game if you haven’t got the latest version (i believe)
Go to Settings > Power & Startup.
From there see if the “Automatically download updates and purchases” option is checked. If it is, then yes the update is downloaded and installed. If it isn’t checked, you should expect to get prompted about the update eventually.
Hold the power button down on your xbox console until it shuts off. Wait a minute and power back on and load up halo. You will be prompted to download it before you see the “343” logo and main menu