I think the EU has common culture of sorts but I think it is very hard to get the Latinate and the Germanic/Nordic cultures to empathise as they are so different. Also, the fact that Europe has, by and large, constantly been at war with each other up until 1945 means that the scars may take a while to heal.

I hope it works but knowing the type of people in the UK, I doubt the UK will ever become a part of it fully. Sadly, in my opinion but there you go.