By the way, watch later is a button in the YouTube player I am seeing and not a drop down menu. The settings (cog wheel) button does open a drop down menu and it works. The functionality of both of them works.
What happens if you use incognito mode (Control + Shift + N) to watch a YouTube video while being signed it?
Also, do you see a console error (press F12 to bring up the Developer Tools, click on Console and see if there are lines with a white X on a red background icon) when that happens?
If you have a firewall anything else that might interfere with your network, try disabling it briefly and check if the issue persists.
Finally, please, see if it works with other browsers on the same system and if you can reproduce this on the same browser in different computer.