Please differentiate between (1) subscriptions to Apple TV Channels inside the Apple TV app, and (2) subscriptions as started outside the Apple TV app, for use in the separate content provider app. These types of subscriptions have different terms and conditions, different perks and limitations, and are not exchangeable as such. BritBox and Acorn TV are available in both ways, with either way possible to have billing through Apple.
Up Next would list movies and TV shows “in progress”. Listings would not show programs for which you have seen the latest episode, until a new episode appears. TV shows may disappear from Up Next if the content provider removes it from their catalog.
BritBox from the separate content provider app would not connect to the TV app for inclusion in Up Next. BritBox from the Apple TV Channels would be supported for Up Next.
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
BritBox app - App Store preview - Apple (US)
Acorn TV app - App Store preview - Apple (US)