Apple One doesn’t include BritBox nor Acorn TV. The video service of the Apple One bundle is Apple TV+, which is its own Channel with its own content (Apple Originals and Apple exclusives).
With the Apple TV app on your smart TV or third party streaming device, you can access:
BritBox in the Apple TV app requires its own subscription, different from the subscription of the same name in the separate BritBox app. Acorn TV in the Apple TV app requires its own subscription, different from the subscription of the same name in the separate Acorn TV app. These types of subscriptions have different terms and conditions, different perks and limitations, and are not exchangeable as such. Choose your preference and stick with that. Apple TV app Channel subscriptions have to be started in the Apple TV app, and need to be viewed in the Apple TV app.
Subscribe to Apple TV channels in the Apple TV app - Apple Support
Apple TV Channels FAQ: Services, pricing, availability, and everything else you need to know - Macworld