My daughter was christened RC, even though my partner and I weren't married. I'm not even a Christian. All it took in our case was one of the parents to be a believere and for both to agree to bring her up as a Catholic. Probably different elsewhere.
I'm a single parent, and I had my twins Christened with my Mum taking on the role of the father. It was the church that my mum and dad were married in, and me and my sister were Christened in, even though we dont live in that area anymore. I dont think it matters, as long as you bring your child up as a Christian.
I can only talk about Church of England as that's the church I am a member of.
Your marital state will have nothing to do with your child's baptism in your own parish church. However if you choose a different parish then you will need the permission of the vicar of your own parish church.
As the C of E is an established church, everybody lives in a parish - you don't have to go to that church to have any of the rights associated with living within that parish.
Just a thought - if you are not actually a church goer have you thought about a service of thanksgiving or a blessing?
Or just your own naming ceremony?