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.
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