I added Apple one so will my family automatically be switched over from my old plan

Make sure Family Sharing is set up first - Family Sharing - Apple

Once you sign up for Apple One then they have access to all those features on their devices. Apple TV+ is in the Apple TV app, Apple Music in the music app, Apple News+ is in the News app, Arcade is in App Store, Fitness+ will be in the Fitness app.

Yes, everything will be switched over automatically. You can add up to 5 family members other than yourself.