I think it would be neater if the time is simply displayed on the top bar itself.
Rationale: The display of UTC time is not related to Achievements in particular, even though it's most used in relation to gaining reputation. The time is related to Stack Exchange as a whole, and should therefore be shown in a place representative of that.
Time is not displayed, when none of the three panels is open:
When I open any of the panels, the time should fade in at the top bar. It should remain shown when I switch between panels. When I close the panel, the time should fade out.
As an advantage, the time could also fade in when the mouse pointer is hovering above it, and fade out when pointer is moved away. Like a spoiler (as shown with fade effects on desktop site):
PS: I felt the term time (as in
UTC TIME 09:20) was redundant, so I favored the format
09:20 UTC in my mock-ups.