J.
In contrast to Kristy's experience, I was raised in a formal religion (Baptist, fundamentalist) and went to a Christian school as well as a church where my grandmother was a deaconness and Sunday School teacher. I was only too happy to move on with my own not-very-religious life! Having it forced upon me so early really made it hard for me to have any appreciation for it. I really don't think you can predict what your children will be like, spiritually. There's something to raising your kids in a loving community, which a lot of people get through their church, but I think it's something you can build in many ways.
Having said that - I'm married to a Catholic and we're half-heartedly raising the kids Catholic (they were baptized, we go to church for the major holidays) and as a non-Catholic I do feel like I'm standing on the outside and excluded a bit. When it comes to church and social activities, I mean - we don't attend the school.
I think it would be worth taking the time to get to know the school and its community and find out how welcome you will feel, because I think it's important to feel a part of your childrens' school. Raised Catholic yourself, it may be pretty easy for you, but in any case I think meeting some parents first is really key in any private school choice.