Unfortunately I can no longer find the actual BBC funeral footage on YouTube or anywhere. Due to licensing I cannot view it on BBC.com either. (I'm in the US.) I was just hoping someone else might have noticed the music at the time of the broadcast and known what it was. Thank you for any clues. My next step will be to contact the BBC directly.
I contacted the BBC and they kindly responded very quickly to my request. The piano piece that accompanied a photo slideshow at the 15 minute mark in the funeral is called Nuvole Bianche by Ludovico Einaudi. It is quite lovely and moving.