(Disclosure, I may receive a small referral fee when you make a purchase through links on this post.)
Chromecast has proven to be a big hit for Android, iOS, and computer users but Windows Phone has been a bit left out. There is limited Chromecast support for Windows Phone meanwhile support for Android and iOS devices have been growing.
That may now change. Windows mobile phone users can keep their hopes up with one app. Tube Cast is an app that allows mobile phone users with Windows OS to cast Youtube to their HDMI screen. The functions are limited as in you cannot pause, go back, or go foward but it is enough for you to select a video and play.
So far it is getting between 4-5 stars in the windows phone app store and the latest reviews are very positive. Give it a spin!