Okay, this blog will cover facts about the gap years in Avatar. The Gaps generally being during the Hundred Years War and Before Korra's discovvery as the Avatar. I will generally address these as they come to me, but for the first, I've come up with the birth of Aang and Katara's firstboorn, Bumi.
Right, now, from the Timeline, we know that during the Search, Aang was 14-15 and there was no Bumi, so that is roughly around 103-104AG. So suffice to say, he wasn't born here.
Tenzin, Aang and Katara's youngest was born in 119AG according to the wiki timeline. So he had to be born sometime between these two dates.
Now, let's take into account basic biology, the gestation period for 1 baby comes to 9-10 months, so let's call it a year at least. So for Katara to get pregnant and have Bumi after the Search and before 119, we have to shave a year for Bumi and a year for Kya before Tenzin. So it now has to be 104-105 to 118.
Now, let's skip ahead. Bumi retires from his position in the United Forces in 171. Now let's assume, that retirement age in Avatar is roughly the same as America, which is about 65 to my understanding. So let's shave that off and that gives us the year 106. So for the sake of argument, let's take another year to narrow that window and make it a bit more solid. Our gap is now 106 to 118. But, as we hear from the siblings themselves, they always refer to things growing up with each other.
With the current maths, that would mean at the earliest, Katara had Bumi at age 18 and he would have been 13 by the time Tenzin was born. As the last of a series of siblings myself, I never really consider myself as having "Grown Up" alongside my eldest siblings with whom I have a 10 year gap, nor would I expect said elder siblings to feel the need to prove their capability against someone 12 years their junior, bender or otherwise. Now this isn't to say that all people follow this kind of psychology, but for the intensity of Bumi's seeming rivalry with his siblings' abilities, I think we need to close that gap some.
Unfortunately, this is where I have to get sketchy by applying more Psychology and scheduling to the gaps in timing. Bumi seemingly feels the need to prove his ability against his siblings despite his lack of bending power, and it's noted that once he was old enough, he joined the military, so likely at 16. At the same time, he and Kya show the same bitterness at not going along on trips with their father as Tenzin did. If Bumi was already away with the military at this time, it's unlikely he would have expected to go at all, so they would've had to have happened before he turned 16 and since we know Tenzin's age, we can use this as a marker. A person's earliest memories tend to be from around 3-4 years old, but one of his trips was to ride the Elephant Koi at Kiyoshi Island. Let's assume that the skill set required for this, he'd at least have to be Meelo's age to pull off, which is 6. So that means Tenzin is 6 in the year 125 before Bumi turns 16, our current gap between these two is 13, which would make Bumi 16 when Tenzin is 3, which is too early, so let's add 2-3 years.
That makes our current estimate, 108-9 to 118, much more managable than before. This currently means that, evenly spaced, there's a 5 year difference between each of the siblings. At this point, we have to take a look at the other two, Kya and Tenzin's psychology.
Tenzin claims that he had to be the responsible one growing up because Kya went off to "Find herself" and Bumi wwas always a bit of a man-child and Kya's refute to this is when their Father died at which point Tenzin was 35, so we can assume that Kya didn't see herself as the responsible one when they were younger, not having raised her older sibling like Katara and feeling comfortable enough with the situation to leave home to find herself. Since Bumi already left at 16, it's unlikely she did before him being younger and 16 already being more or less the earliest socially accceptable age to fly the nest, but there was enough of a gap between her leaving home and Tenzin doing so to feel he had to be responsible, so to me this makes a bit more of a defined gap between her and Tenzin than her and Bumi, so let's add another year for the gap between Tenzin and Kya, making her at the least, two years older than her youngest sibling.
108-9 to 117
Honestly, given the closeness of the siblings, I'd be inclined to bring this down to about 115 for Bumi, 117 for Kya and then 119 is set as Tenzin, but unfortunately given what evidence we have and the best I can say for their respective psyches, the above gap is the best I can do without basically saying I'm guessing.
I will follow with this though. It's likely that King Bumi of Omashu unfortunately passed from the Avatar World sometime shortly before Bumi's birth. Not being a girl, Kya would be off the table, but why else do you name your first born after your friend as opposed to the man who raised you as they did with their following children? The only reason I can think of, is that it was in honour of a friend who left that world at about the same time, making King Bumi's final age between 120-128 give or take. Not bad for the one who took the long way around.