I checked my account now and there was no progress for me in the main PS4 game. But then I found out that the progress you achieve in the App campaign is not the progress which should carry over to the main game. Instead, inside the app you should got to ‘Connect to Evolve’ (the place inside the app where you can watch replays) and there you should see a new third button which only appears when you have bought and connected you Evolve game on PS4. There you can spend the ingame currency you earned to progess your hunters from the main game.
For the PS4 DLCs. Turns out, there was also now download for me concerning the Skins etc which you get as a Preorder Bonus. Only think i could download where the magma skins because of the season pass which was included in the bigger PSN Edition of the game. Anyway, i still had all the skins an bonuses activated inside the game, even if I could not see them in the PSN Store or somewhere else. Next time you are on the character selection screen, check out goliath and there should be an option right under his name which lets you change skins