Classic Shell only adds a Start menu to Windows 10. It should not affect the taskbar clock functionality in any way. What do you mean no access to clock controls? Does clicking the clock not show you the time zones?
Btw in Windows 10, the clock UI got rewritten and is quite buggy. Maybe it will work if you revert to the old style clock:
http://winaero.com/blog/there-is-a-new-pane-for-date-and-time-in-windows-10-build-9926/ Set the value of UseWin32TrayClockExperience to 1 to enable the old style clock.