Conor's father moved to America with his new wife six years before A Monster Calls takes place. He comes to visit Conor for a few days wile Conor's mom is in the hospital, but then he has to rush back to America because his new baby is sick. Dude's got priorities, we guess.
Conor wants to go live with him in the U.S. instead of living with his grandmother after his mom dies, but his dad says, "We barely have room for the three of us, Con. Your grandma has a lot more money and space than we do." (13.39) By "the three of us," he means his new wife and child.
In other words, even though his dad is still dad-ish enough to come back to England in an emergency, he's not willing to allow Conor back into his day-to-day life. Whether or not it's actually true, his new family seems to be more important than his old one. And his new baby seems more important than his old son. Even before he knows the baby is sick, he only plans to stay a few days, telling Conor, "Americans don't get much holiday" (14.7). This is one of his many excuses for not calling and visiting as much as he should; others include his new wife's frequent migraines.
He's not a bad guy, Conor's dad. He's just not going to win any father-of-the-year awards, either. But hey, as with most people in A Monster Calls, it's complicated.