The 24-year-old's Scottish fiancée Clair Meek gave birth to a baby boy at Crosshouse Hospital in Kilmarnock.
"Clair was determined she was going to have their baby in Scotland," The Sun quotes a friend as saying.
"They are the happiest people in the world. They are so thrilled and proud. The two of them are so in love and now they have a beautiful baby son."
Best known for his role as Jay in The Inbetweeners, he first revealed details of his fiancée's pregnancy in June earlier this year.
Buckley has been dating 23-year-old model Meek for over a year.
> Read Digital Spy's recent interview with James Buckley