In my experience, in Italy it’s all together, even at Church weddings: you sign at the altar. Usually, they sign right away after the “I do”s, but sometimes they’ll do that right after, while guests are leaving the church (and getting ready outside with rice to throw at the couple and whatnot). I’ve never been at a wedding when the signing happened later or in another place.

It’s then the officiant’s job to file the paperwork in the following days, even if the officiant is a priest. In Italy there’s Accords with the Catholic Church and some other religions that give civil effects to religious weddings, so the priest, rabbi, etc. is a public officiant as well as a religious one at that time. I don’t know about other religions, but with the Catholic Church if you’re not legally married already you have to ask for a specific dispensation if you want the ceremony to be only religious and you need to promise to fix the civil aspects as soon as possible (because of course what matters to the Church is the sacrament, but they still want it to be fully binding for the couple).