As simple as the title says, as far as I am aware, you can't actually use the card backs from the non main nations even if you have them as allies in your deck.
What I mean for unused card backs are those that appear when you open card packs to identify them easily.
While really simple, it is quite a wasted opportunity to have more options, specially for newer players.
The concept would be quite simple, you'll create a deck and choose any mayor nation, and then if you pick "Finland", "Italy", "France" or "Poland", when you want to customize it and change its card back, you'll have besides the vanilla back or purchased, you could also choose the ally nation's design.
Doing this would obviously reveal which ally cards you may have and ruin the surprise element (Which I think is not that relevant)
And last but not least it would not apply for other nations card backs, lets say you have another mayor power as an ally, you still wouldn't be able to use its cardback, it would be too confusing.
Edit: (The image used is a visual error, the deck I've used was russian/german, didn't even had a polish exile card for that cardback to display)