My wedding 26 years ago was a full Catholic ceremony blessed by the Church. The presiding priest was the only one we had to talk to about it. I was neither religious nor baptized at the time. The only other rule was that at least one of the witnesses had to be Catholic.
She did remind me that it would be a trivial matter for her to get it annulled if she wanted. I went through RCIA a couple years later. Now it would be harder for her.
Two more data points on annulments: I know someone who tried to fight an annulment. It went through anyway because he had a child with one of his mistresses. After the annulment he could have gotten married in the church again.
Also, my sister got her first marriage annulled even though neither were Catholic at the time. She converted in her late 30s and wanted the opportunity if she met someone. She got married a while later in a civil ceremony with a judge and only two witnesses. No idea what the priest thought of it but she continues to work in lay ministry.