Just some further thoughts then:
An internet radio (which gets its signal from your router, rather than across the airwaves) might be a bit of an extravagance if you only want to listen to Radio 4 but, if you're interested in hearing lots of other stations as well, it's a possibility that could be considered.
I listen to a great deal of radio via my computer (especially the 24-hour British Comedy Channel:
http://rokradio.com/ ) but I'm not pinned down next to it. That's because, last Christmas, I treated myself to a pair of Sony wireless headphones. I simply plug the base unit's jack plug into my computer's headphone socket and wander around the house listening to great comedy. (The headphones are incredibly comfortable and the built-in rechargeable battery lasts for up to 14 hours). So if you've got ANY radio or computer that can receive what you want to hear (irrespective of its location in your house) you can listen all around the house.