A few people seem to have this problem in the OT, but I think most have found a fix for it after they brought it up in the thread and were helped by Rushy from Evolution or they tried the tricks other people have tried which has made the DLC work.
1) For some a reboot of the game or PS4 helped (somehow it didn't register in game that the DLC was installed until a reboot)
2) are the game and DLC from the same PS Store? if not, there might be a problem and you need to try to download the DLC from the same region's store
3) is your user account from the same region as the game and/or DLC? if, say, your game is from the US Store but you use an European account to play the game, make sure both accounts (the one you bought the game with & the one you use to play the game) have your PS4 set as their Primary consoles. Or make sure that your PS4 is the primary console no matter if the DLC & game are from the same store.
4) if none of these work, then you might have to re-download the game & DLCs. Try to download the DLC differently from how you did before. So if you previously downloaded it from the PS Store, perhaps now try to do it through the game.