It is rather strange that bloodbending became known to the public by the time of Yakone's trial, considering that the dark art was not known to many around the time of The Last Airbender. I find it rather unlikely that Hama told someone about bloodbending before Katara as it is probably not too likely that a waterbender stumbles across some random Fire Nation village she was residing in. Also, Hama probably found Katara's origins from the Southern Water Tribe as more incentive to pass on the skill to her. If this is the case, then Katara certainly had to be the only one who Hama revealed the existence of bloodbending to, as she was the only waterbender from the Southern Water Tribe in existence since Hama was captured decades ago.
So the way I see it, there are only two possible ways that bloodbending became known to the public, as Katara and the rest of Team Avatar would have kept its existence a secret due to its dangerous potential and those who would use it. First is the residents of the village that Hama terrorized, who could have told their friends in other villages and towns after they realized how Hama controlled her victims. That news would spread through the whole nation eventually, and after the unification of the Fire Nation with the rest of the world, through the other two nations as well. The other possibility is the Southern Raiders commander who Katara and Zuko mistook for Yon Rha. He could have told his fellow commanders and the rest of the Fire Nation military how his body was controlled by a waterbender.
Whether either possibility occurred or not, evidently, bloodbending became well-known not too long after the series. It had to have occurred well before Yakone's trial too, because obviously, bloodbending had to have been an issue prior to it being outlawed. Yakone also claimed that his family had a line of powerful bloodbenders, indicating that, if there was truth in that statement, his parents at the very least had to have known about bloodbending as well. With these known facts, it can be presumed that bloodbending became well-known very soon after the conclusion of the series, within a decade at the very least.
Something I thought about was, since the upcoming graphic novel trilogy centers around the two Water Tribes, could it possibly reveal more about how bloodbending became a well-known skill despite the very few who knew about it when it started (and the likely assumption that most who knew would not have wanted for the knowledge of its existence to spread)? Since the graphic novels are addressing unanswered questions from either series as one of its main roles, the knowledge of bloodbending ought to be covered eventually. Is it possible that the mysterious intentions of Malina, who may be the main villain of the trilogy, has to do with bloodbending? If so, she may have a connection to Hama. Here are two theories I came up with.
First theory is that Malina is the daughter of Hama. Sometime between her escape from prison and meeting Team Avatar in the village, Hama had a daughter (possibly from someone from the Northern Water Tribe, as Malina is said to be from the north). While raising her, Hama taught Malina how to bloodbend. For unknown reasons, Hama then went back to the Fire Nation, possibly due to her desire for revenge (which wasn't satiated by her new life, something that commonly occurs in Avatar villains like Yakone). Meaning that Katara wasn't the first person Hama passed on her bloodbending knowledge to. But now, during North and South, Malina is uniting the two tribes with a hidden intention that involves bloodbending. An intention that would result in bloodbending becoming known to the public.
Second theory is that Hama escaped from her incarceration after being defeated by Katara (I mean, it couldn't have been that hard since Hama could have just bloodbent one of her guards to release her like she did before, unless the villagers took special precautions). Since then, she searched for another waterbender to pass on the knowledge to. One of them may have been Malina, which leads up to the second half of my first theory. But even if that's not the case, if Hama indeed escaped after The Puppetmaster, then it would not have been that hard to make bloodbending known in a short time.
One more thing: is it plausible that Yakone is related to Hama? Yakone's claim that he came from a line of bloodbenders seems really dubious and inconsistent, as bloodbending could not have been known to those other than Hama and Team Avatar at any time preceding his parents (Yakone is around the same age as the original Team Avatar and so during the time that bloodbending became known, the only generation above Yakone that could have learned bloodbending is the one directly above him). But just from the way that Yakone described his heritage, it is indicative that bloodbending has been practiced in his family for a long time (assuming that he didn't exaggerate or lie about anything to make his sons have an easier time being convinced). Something that I thought of that could be possible is that Yakone is Hama's grandson. Between the time that Hama escaped from the Southern Raiders prison and her second incarceration, she may have had a child (like stated in my first theory). Although it could be Malina, the indication that Malina could be Unalaq's wife's mother (due to their shared name) could be evidence against this as there has been no indication in the series that Unalaq's family is related to Yakone's. Needless to say, Hama would have trained her child bloodbending. That child would then be the parent of Yakone. The time gap between Hama and Yakone supports this as possible, as Hama and Kanna were around the same age and Kanna's grandchildren are around the same age as Yakone. If this is the case, then Yakone's claim about his bloodline would make a lot more sense as his grandmother is the one who actually discovered bloodbending and it has been passed on for two generations (and Yakone is passing it on to a third)