I can confirm that for an active Sim who did not just have a birthday, it counts up at midnight.
The green bar represents days, I'm pretty sure, so it shouldn't show green until 1 has elapsed. The day of a birthday it wouldn't have filled in for their previous lifestage until that midnight, so, despite you using the cake to age them up earlier, logic seems to imply that it should not get any green at all at midnight on the day of their birthday because that day was part of their now-passed lifestage. It should increment for the first time at midnight the following day.
I keep forgetting to pay close enough attention for 2 nights in a row after a birthday to be sure how it works though.