YouTube at double speed on mobile
Updated answer for those who still prefer to watch in Safari

In the iOS Shortcuts app you can find a shortcut that will change the speed of any video it finds on the web page you're on when you run it. You run it from the share sheet in Safari.

Activate it in the Shortcuts app in Gallery → Javascript Shortcuts → Change video speed.

Visit a Youtube video in Safari and use the share sheet to run the newly added shortcut. It will ask you what speed to play the video at.



Original answer


On the video's mobile Youtube page, tap the menu button at the top left (or right). Scroll to the bottom and switch
to Desktop view.*


 


Change the speed controls just like you would on your computer.
Play the video by tapping it and it will use the new speed settings. 


*If you're in YouTube desktop mode but still don't see the gear icon then tap Safari's action/share button and request desktop version from there as well. The request desktop icon looks like a monitor/iMac.



(I recently learned that you can also tap and hold Safari's refresh button to request the desktop version of a page)

If you lock the screen Safari will pause playback, but it can be resumed either from the lock screen or your headset to allow audio only.