Processing takes a while esp with Covid. It’ll kick in once you give them the birth certificate or proof of birth from the hospital. They have to check in with his employer and make sure he meets the requirements as well. You have to work for so long and have so many hours which takes time to get all of the documents together. His employer also has to meet requirements.
Google paternity leave and your state, should be able to apply online. Not sure of their turn around time with applications. Should file 6-8 weeks before leave is needed and then it’ll kick in with proof of birth.