Is there a way to make Siri play BBC radio stations?
Long story short I don't think there is an official way at the moment.

Third-party developers like the BBC can add support for Siri to their apps now, but that is only allowed for certain use cases, and streaming radio stations doesn't seem to fit into any of them. The closest option is the CarPlay option, but that would only be useful to you while you were in the car and if your car supported CarPlay. People have asked the BBC in the past whether they plan to support CarPlay, but the BBC didn't respond.

List of allowed use cases for SiriKit from Apple's Developer site:


VoIP Calling - Initiate calls and search the user's call history.
Messaging - Send messages and search the user's received messages.
Payments - Send payments between users or pay bills.
Lists and Notes - Create and manage notes and to-do list items.
Visual Codes - Convey contact and payment information using Quick Response (QR) codes.
Photos - Search for and display photos.
Workouts - Start, end, and manage fitness routines.
Ride Booking - Book rides and report their status.
Car Commands - Manage vehicle door locks and get the vehicle's status.
CarPlay - Interact with a vehicle's CarPlay system.
Restaurant Reservations - Create and manage restaurant reservations with help from the Maps app.