So your timeline stretches pretty far, ranging from the Age of Myth to the time of the Legion of Superheroes. And time travel has come up now in the histories of Donna Troy and the Captain Marvel moniker. Presumably characters like Booster Gold and Reverse Flash exist, and the mention of UNIT in the original post implies that Doctor Who exists, meaning time travel is pretty firmly entrenched in the BronzeRealms. What forms does this take? Are there any unique rules to time travel in the BronzeRealms? Are there groups that deal with time travel violations like the Time Variance Authority or Rip Hunter’s Time Masters? Are there any other regularly time traveling superheroes, supervillains, or teams? Are there threats that span the timestream like whatever it is Hal Jordan did in Zero Hour? …This might need to be broken into several posts, honestly.
Let's talk about the second part of this first, and maybe I'll come back for the rest in a Part 2 or Part 3 later. So for now, authority figures who try to enforce some conduct of time travel as their jurisdiction, and/or try to preserve the timeline from negative/excessive interference.
There aren't as many as you might think by looking at how many things the BronzeRealms covers! That's bc I've consolidated a lot of them. Rip Hunter, Jack Harkness, and Bruce G. Braxton of the USS Relativity all went rogue from the same organization. They were all Captains of Federation Timeships from the Federation Time Agency in the 3360s through 3540s. Colloquially, Time Agents.
Time Agents are by far the most likely temporal enforcers or protectors for any given would-be time criminal to have to evade. It was a whole fleet that operated basically unchallenged for almost 200 years, and that's just before the Temporal Cold War started taking up most of their schedule. They were technically still active as far as the 3620s, albeit stretched much, much thinner.
The Legion of Superheroes, formed in 4037, has a much lighter touch than the Agents, but will step in when it's most needed. And when they do, they are a force to be reckoned with. And when they're done, few people will even know there was a crisis at all. In the timeline of the BronzeRealms, they are more or less the "Happily Ever After".
The Legion of Doom is the anti-Legion of Superheroes, so despite being a little off-topic I'll bring them up anyway. A coalition of supervillains trying to subvert the timeline for their own personal gain. Vandal Savage, Reverse Flash, Per Degaton, Tempus, Epoch, Abra Kadabra, Chronos, Nimrod, various incarnations of Kang (at different times), and others. The exact roster varies from scheme to scheme due to constant infighting and directly conflicted end-goals as to how exactly they want to change the timelines.
Once, there were Time Lords. That was before the Great Time Wars (sometimes called the Last Great Time War, but... that proved kind of aspirational, kind of like World War 1 being called "the war to end all wars"). However, despite the Time Lords now being sealed inside a painting where they can no longer interfere with the timestream, time travel being time travel, from time to time someone will still encounter pre-War Time Lords who accidentally slipped through the Time Lock and will try to enforce their own long-forgotten law.
The Doctor is still here. That painting is in their study. The Doctor may be only one person (with many faces), and isn't even looking for trouble most of the time, but, they have nothing but time to happen across it and set things right. By pure abyss of age, The Doctor's presence approaches rivaling the Legion's.
Bunnyx is also only one person. But she may wield more power than anyone else here, possibly put together. Her command over the Time Vortex is so deep she can actually just shut down nearly any technological method of time travel, permanently, with a wave of her hand. However, she has sworn to non-interference except in cases of galactic extinction or greater. Her perspective even allows her to see some yet-to-come timestream alterations before they take effect, travel through most Time Locks, or revisit defunct timestreams with less (not none, but less) risk of destabilization or leakage into the current timestream.
The Legends of Tomorrow, founded by Rip Hunter, but very early taken over by White Canary (Sarah Lance), are a team who travel on the stolen Federation Timeship USS Waverider. Despite this shaky, criminal start, their relationship with Agents has progressed into more rivalry than hostility, thanks to many years of doing more good than harm.
The TVA is also a thing that was founded by Kang in one period of his life, and, because it failed, there are multiple versions of the TVA that persist after Kang's departure, all from different timelines that responded differently to the resulting power vacuum, all trying to enforce their own interests and prove themselves as the "true" TVA. You might even get TVA Hunters from competing TVAs responding to the same incidents, or responding to each other. Some of them are reformed, most are diverse shades of bad, and they all look identical on first glance. It's generally a mess and the reason they're not more of a problem for everyone else is because they kind of keep each other busy.
Kang will return (in his own post).