I spoke too soon, it works sometimes by typing quickly the UBS password in the UBS Access app and going back to the UBS key 4 app but it nearly never works. I have to try multiple times... It's extremely annoying :/
Basically :
I open the UBS key4 app -> Log in with UBS Access -> UBS access opens -> Vanadium open with the page i linked above. As a result, I can't access my banking account :(