Your answer needs to come from someone who deals with annulments on a consistent basis. Having taught RCIA and this question, there are so many variables that from what you’ve written, no one can give you a good answer. For one instance - you say he goes to another church. Is it Catholic or another Christian denomination? Makes a big difference in the answer to your question.
For instance, if one or both parties was a baptized Roman Catholic and married outside the church without a dispensation, it is what is called an “error in form” and no lengthy inquiry is needed. It was “de facto” not a sacremental union. And remember, a decree of nullity (an anullment) only affects the nature of the sacrament of matrimony and whether it was properly entered into. An annullment is not synonymous with a civil divorce.
If however, one or both parties were baptized Roman Catholics and all the proper steps were taken to make the marriage a sacrament, then if he objects, a length inquiry as to all the circumstances is necessary. He would have a right to object, although I can’t see how he would, given that he has already remarried. And, an annullment has no bearing on the effects of a civil divorce (i.e. it doesn’t affect alimony, child support payments, division of marital property, etc - those are affairs of the state, not the church)
You don’t say if either you or your ex are Catholic. One thing though, if he is Catholic and married you in the church, his second marriage is not viewed as “valid” by the Church. And no, if he is Catholic and again married outside the church, regardless of the disposition of your petition, his marriage would not be viewed as a valid sacrament.
I would suggest though, that from your standpoint, his second marriage, at least as it affects you, your petition for a decree of nullity, and your status in the Church, has no bearing on your status.
And again, seek the advise of a priest or whoever is handling your cause from the church’s standpoint. There is so much that affects the “annullment” process, that from the information you provided, it is impossible for any of us to know the proper answer.
My two cents.