The problem can be solved by enabling "input indicator" Customize notification icons -> Turn system icons on or off

The quirk here is that once enabled you get the indicator but it may stuck at single language, i.e. switching does not work even though additional keyboard added. I solved it by adding and then removing extra keyboard layout:

User has to be member of the "Administrators", to access Control Panels!

Provided the above is met, to open the "Language" Control Panel: In Explorer (or Run menu, [WinFlag+R]) paste this path

Control Panel\All Control Panel Items\Language

There is a button above the language list "Add Language". Click it to add one. As soon as you have more than one language listed, in System tray appears the indicator, from which they could be changed.

I think we can close this question as I solved this. This was some issue with expired Windows Insider build which was unable to receive any new updates. I tried to add new Microsoft Account and get the same issue, then checked build (it was expired) - I installed RTM build and language bar working OK now. So I think this was an issue specific to this early build.