I'm am male, over 40, have children and used to play a lot of videogames.
The problem is probably threefold and probably needs three separate solutions.
He might have a mild video game addiction. Talk to him about this. Ask him what he thinks. See if you can talk him into going cold turkey for a week. It would be best if it included the phone. When he does he needs to keep a good eye on how he feels. Does he get cravings, irrationally frustrated or angry. Look for excuses to use a screen. For this to work it needs to be a group effort that you guys test out together. It won't work if you make him, because then he'll just lie about how he feels when not playing and tell you he's fine. He needs to want to try and examine the effects it has on him. If he finds out he has a problem work on that. He can limit his own time, look for professional help. Deal with it as an addiction.
Having a baby is stressful. Also for the father. In some cases more so because he feels responsibility but no way to participate directly with the health of the baby. He can help the mother and help out with making money for the family. But he lacks the tits to actually do the important stuff. So he might just be trying to deal with that stress. Talk about this. Most new fathers just get ignored when the baby arrives and they can freak out a bit while internalising it all. They need help but feel like they need to be the strong one. He might just be running to games and screens to cope with the stress. Talk to your partner about this. Ask him how he's feeling about these issues. Does he need help? Can he get it from an older brother, father or some other person with experience.
Because of that it's hard as a father to bond with the baby that early. You can hold it, but you can't give it anything it needs. He might not feel like he can bond and therefore feels left out. So feeling left out and stressed and possibly slightly addicted he's just doing what makes him feel better to drown his sorrows in not being able to bond. You feel like you should bond with the baby, but you can't and it causes guilt. He might be trying to deal with guilt and using screens to do it. A solution here could be that he gets a job that only he does with the child. Bathing it is a good one. It's fun for him and the baby. It's cute as hell and it can be done every day. He can be responsible for that every evening after dinner for example. It's his and make sure you don't accidentally take it over.
So...
It might be addiction. Talk to him about it.
It might be stress. Talk to him about it.
It might be guilt. Talk to him about it.
Don't tell him what to do, just ask him how he feels and offer support.
PS. Dungeon and Dragons in not a video game, it's a boardgame with a strong social aspect. Does he play that in person with his mates ? If not, then maybr that could be something he could do as a social thing and stress release.