I've tried googling and can't find any recent answers on this. As a bit of background I bought the base game and played the first scenario through and loved it so much I ordered the expansion pack.

Do I need to restart the campaign after adding the expansion to my collection for them to show up? Or will it work when the bosses appear in my current game at the right time?

The bosses are not changed by buying the Villains of Eriador pack. The pack provided physical representations of bosses where the game was using substitutes before. Like Gulgatar was an orc marauder, but with the Villains of Eriador pack you have a figure for him and his icon will change to represent that now.

You don't need to start over. The game will give you the new cards if you had earned them prior. I'm not sure the app will give you a notice, just activate the pack on your device in the Collection section of the main menu and take a look at the group inventory.