The time between ranks is equal to dan rank in the ATA, aka 1st to 2nd is one year minimum, 2nd to 3rd is 2 etc.
3 years for 2nd though, that'd be a quick pace for an adult who trained regularly.... for a young child at least in our school, we're looking 1 stripe at a time (1/2 gup), and only testing if they're ready to. With 2 month testings starting at 9th white... 38 months if nothing goes amiss. But then it's exceedingly rare to start anyone under 5 at ours as well.
I guess it varies by school, hopefully the little one will continue on and grow into her rank.
CF'er, you'll find a huge variety in the ATA, some schools are much more traditional, some are the shorter classes etc. It all depends on how the instructor was brought up.
I was of the era of knuckle push-ups in the gravel, running barefoot around the block a few times with the class, making the windows sweat etc era.
My take on it is this, affect the bubble around you. Basically I can only improve those I come in contact with etc.
Btw, my own children are 4 and 8, and I'm just now considering starting them learning. With the young one I think I'll 'homeschool' the TKD for a while, simply start installing the basics with no ranks etc.
6 year old 2nd... wow. We used to have a minimum age of 18 for 2nd, 21 for 3rd etc, unless my memory betrays me.