I've just bought Hearthfire, Dawnguard and Dragonborn. The money went from my account and all 3 have downloaded however none of them appear to have installed. Is there any way that I can check whether they actually have installed and, if not, what do I have to do to install them?

Click Data Files in the launcher and make sure they're checked.

Thanks for the quick reply but where do I find Data Files?

As noted, in the Skyrim launcher - Data Files is the third option down.

Duuh... thanks, I was looking in the Steam launcher (?). Okay. They are all there and all are checked. So I guess I just load the game up and see what happens? Will I get any notification from within the game?

There's no direct notification.

One way to check is to use a forge and try to make arrows, an option added by DG.