Clash Royale sometimes broadcasts interesting battles from each arena to the TV Royale listings. If a particular battle makes it to TV Royale, are the participating players informed that their battle was featured?
Yes, you and your whole clan will be notified via push notifications that a battle including a clanmate has been featured.
From the February 2, 2016 patch notes:
Push Notifications: New "A Clanmate's battle is now featured in TV Royale!" notification