Hello,
I have never listened to BBC Radio (or any other radio station) with my HomePod mini so I am not sure this will work but it does allow me to listen to music (from my iPhone) then use my AirPods with my iPhone for another app. Since you cannot control BBC Radio with a HomePod mini this may not work at all.
You would want to start BBC Radio playing on your iPhone, like you are now, but do not AirPlay to your HomePod mini. Instead, Handoff playback to your HomePod mini my holding your iPhone to the top of the HomePod mini. After the audio has moved to the HomePod mini you should be able to connect your AirPods to your iPhone to have any audio from your iPhone rout to the AirPods.
Play audio on HomePod using an iOS or iPadOS device - Apple Support
"Hand off audio to HomePod
You and members of your home can transfer audio from your iPhone to your HomePod to keep playing from where you left off.
To change who can hand off audio to HomePod, see “Allow others to control audio on HomePod,” below.
Note: To hand off audio, your device must have Bluetooth turned on and be on the same Wi-Fi network as HomePod.
- On your iPhone, go to Settings > General > AirPlay & Handoff, then make sure Transfer to HomePod is turned on.
- Hold your iPhone near the top of HomePod.
To move audio from HomePod to your iOS device, hold your iPhone near the top of HomePod, then tap Transfer to iPhone. If Ultra Wide Band isn’t supported, audio will move automatically when you hold your iPhone near the top of HomePod."
3rd Party link to WhatsApp's Help Center “How to play voice messages”
I hope this helps.
