There is some problem with youtube servers in India that's why some users are experiencing this issue.
P.s I am also from India but I am able to play it on desktop,mobile and youtube apps this problem is for some users only.
Check the Mobile memory, if u have filled it almost it cannot play YouTube videos!! As the video needs buffering to be done before and that data has to be stored!
This prob is occured due to dot guidelines
Of blocking some illegal sites
But this process also affected YouTube bcoz these guys also blocked googlevideos.com
So hav patience it will be fixed soon
Or if u want to stream YouTube videos now then u hav to change ur DNS to google dns