STEP 1 - Go to the contest's web page you want to visit:
(And so on...)
STEP 2 - Visit the "Photo & Video" tab and choose "Web TV" (for eg. http://www.junioreurovision.tv/page/webtv)
(on Young Musicians web page, Web TV is at the tab on top of the page)
STEP 3 - You will be required to install the Octoshape plugin (it's the only way to watch.)
For more info about Octoshape click here.
(It launches every time you start your computer, you can uninstall whenever you want, just search for "Octoshape Streaming Services" on your pc)
STEP 4 - After the installation, if they're broadcasting something live, just click on the specifc link!
And there you have it. Have fun!
(I think at the moment, it's not possible to watch on mobile devices, because it needs Flash Player)