Did you make sure to drag the second ring manually into the slot instead of simply pressing "E" to equip? Pressing "E" only will default to replacing only the 1st ring slot.

Your video suggests that something about the setup is not working correctly for you, but I cannot reproduce the issue even copying your setups as far as I can see, including the class (tested both on live and PTS). I can also see Nirn working fine with only five pieces (of varying slots) in logs from this update. It does not seem to be a general problem with the set or the set/class combo.

I would suggest:

-re-seating your rings and weapons (I see you already did that for the arm cops and amulet)

-barswapping to the back bar and then again to the front so the game updates its understanding of what's equipped
(I know it showed the fifth piece as lit up/active in the tooltip when only five pieces were on, but it didn't proc the set as it seems it should have in that case. You might also try looking at the Info page of CMX and/or using the add-on Currently Equipped to see if it recognizes all five pieces properly.)

-logging out and then back in on that character - perhaps remove the set first to try to clear up whatever's bugging out

-restarting the game

-filing a /bug report with your video linked (for the devs' reference) but also a /help ticket so someone can look at your character specifically to see what is happening and get back to you