What are the various ways to skip a song on youtube on 3rd generation Apple TV using the iOS remote app or Apple Watch remote app
On iPhone

If a song is playing then go to details and you can press the two arrows at the bottom to fast forward/skip a song or you can tap the 10 icons to skip back or forward 10 seconds:



From this Apple Support article:


  Fast-forward: Tap the 10 icon to skip ten seconds ahead in a video. Or tap Fast forward 
  
  to fast-forward in a song.
  
  Rewind: Tap the 10 icon to skip ten seconds back in a video. Or tap
  Rewind to rewind in a song.


On the Apple Watch

Tap the two arrows to skip/fast-forward or rewind the song:



From this Apple Support article:


  To play or pause your music, or to go to the previous or next track, use the controls in Remote.