It's not difficult to get a US passport for your son without bio dad's consent. Simply fill out this form
http://www.state.gov/documents/organization/80106.pdf and write an explanation under the "special circumstances" section about how bio dad is somewhere in N. Ireland and you no longer have contact with him, he's never been a part of the child's life, etc. and submit that along with the passport application.
Someone else can give you more info on this, but I think as far as the visa goes, once you have the US passport for the baby, you would apply for a dependent work visa. Again, I would think you'd have to supply a statement that the bio dad's whereabouts are unknown.
Because the father's name is on the birth certificate, you do not have sole custody unless you get paperwork from a judge declaring this. You can still get the passport without this paperwork as I've already mentioned, but just so you realize that from a legal standpoint, you do not have "sole custody". You will always need bio dad's consent or will need to present a statement where you indicate the bio dad is absent from the child's life. In the future, you might want to look into step parent adoption. This way, you can remove bio dad's name from the birth certificate and replace it with your husband's.
Of course, you can skip the visa altogether and go for British citizenship for the child, but this requires full cooperation from the ex. Only you can decide if contact with him is worth it.
