I can tell you my experience because, just from the basic information it sounds very similar to mine.
I wanted to join the Catholic Church. My husband was a cradle Catholic, not a strictly practising Catholic, and we were “married” outside the Church. I was married before and had received a civil divorce. My first husband was not Catholic. The first night of RCIA, our two priest were there and one of the items was on annulment. I made an appointment and then went to discuss it with one of the priest. The annulment process can be very long and complex, so the sooner you get started the better. It took me ten months to receive my annulment, all the while I attended Mass (could not participate in the Eurcharist) and RCIA (Our children were baptised during this time). I was not able to fully participate in the Easter Vigil when the rest of the RCIA was received into the Church. When my annulment was declared, my husband and I were married in a very short ceremony prior to Mass. During Mass I was received into the Church, and afterwards we had a great big celebration, complete with tears and hugs.
Annulments use to take years, but now only take a matter of months, usually ten, and that is when you stay on top of it, the previous spouse is willing. Some people I knew, they were trying for years to get an annulment mainly because their previous spouse would cause trouble. I was really surprise my ex agreed and see it as a true miracle.