I've just bought and downloaded the Dead Money, Honest Hearts, Old World Blues, and Lonesome Road DLCs for FNV. When I started playing the game from a previous save, nothing seems to have changed at all. There's no new quests, no notification or anything. I go back to the main menu and select 'view downloadable content' and there's nothing there.

So what's happened is, I bought the DLCs, they've downloaded, but the game doesn't seem to realise they're there and I can't access them.

Check if they are checked in Launcher. Also you can just create a .nam file with the name of the file you want to launch, e. g. deadmoney.nam.