How about the copyright people, the CCLI, what do they have to say?

CCLI is the Christian Copyright and Licensing organisation. Churches in the UK and elsewhere will pay CCLI for a license that enables a church to use media – eg projecting song lyrics requires a license.

Here us what CCLI say on their website regarding streaming video from YouTube:

We recommend that you obtain direct permission from the owner of the video. YouTube's terms state their site is for private use only.
CCLI’s recommendation on using video from YouTube

We recommend that you obtain direct permission from the owner of the video. YouTube’s terms state their site is for private use only.

I also emailed them, and they added:
“We always advise that churches use caution when using sites like YouTube, as a lot of the content has been put on the site without the copyright owners permission. It is certainly not something that CCLI could cover your church to do.” October 2013