Who the hell put Adrian Veidt there?
The show hasn't offered many clues as to how Veidt managed to find himself banished to a moon two planets away from Earth. Three people are likely to have been involved with him getting into his current predicament: himself, Dr. Manhattan, and Lady Trieu. Let's tackle them in order.
Veidt could have done it to himself, perhaps on purpose, perhaps by accident. He's pretty old. Per a document in the Peteypedia, he would have turned 80 a couple of months before the present-day events depicted on the HBO series. Maybe he decided he wanted to retire and get completely away from humanity? If you recall, Veidt's scientists created teleportation technology in the lead-up to dropping the squid on Manhattan (and you see teleportation devices in this episode when the 7th Kavalry kidnaps Looking Glass), and they also cloned animals, so he could conceivably have sent himself out into the stars to live by himself for the rest of his days, with only clones for company.
One point in favor of this theory is that Veidt has it pretty great: a giant mansion on a piece of green land big enough for a herd of bison to roam in, with all kinds of technology at his disposal to while away his days tinkering around with, a lake full of clone babies in crab pots. I mean, that's the life! If you were imprisoning someone, wouldn't you want the imprisonment itself to be a punishment? But none of this solves how he could survive on a Jupiter moon without some help from a certain blue being.
Also confusing matters is that, according to the report, Veidt was declared missing in 2012, the same year that Trieu Industries purchased his remaining businesses. As of 2017, Trieu Industries also controlled Veidt's estate, and the man known as Ozymandias was declared "presumed dead" by multiple world governments and Trieu Industries in 2019. It's doubtful that Veidt knows anything about that, given that he's likely been isolated two planets away, likely for years. In fact, it seems certain that he has been duped by whoever put him in his current location. But who put him there?
Dr. Manhattan is the only being in the known Watchmen universe with godlike powers. In the comic books, he made it possible for Laurie Blake to breathe on Mars by creating a bubble around her. When Veidt catapults skyward, he reaches a point where the atmosphere changes from habitable to unhabitable -- in other words, he's pushed through a bubble that someone created into the surface of a moon, and the best guess about who could have created such a bubble is Dr. Manhattan. But did he put Veidt in the bubble? If Dr. Manhattan didn't imprison Veidt after the events in the comic book, what could Irons' character have done to make Dr. Manhattan concerned enough to do that to him this time? Also, Veidt's note -- "SAVE ME D" -- seems to have been cut off, and the most likely intended recipient (the "D") of the message would be Dr. Manhattan. Why would he think Dr. Manhattan would save him if Dr. Manhattan had put Veidt there in the first place?
That brings us to Lady Trieu. As far as I can tell, she doesn't have the power or technology to send someone to outer space, let alone create a habitable bubble utopia on a distant moon. Remember, though, that previous episodes have indicated that Veidt willingly accepted being sent to the Europa utopia, as part of an agreement that he seems to have violated, per the masked warden. Could Lady Trieu have double-crossed him? Who would have the intellect to double-cross the self-proclaimed smartest person on Earth? Trieu seems pretty smart. She also now owns his businesses and controls his estate, in a still unexplained turn of events, and the timeline of that transfer of ownership seems to match up with Veidt's disappearance seven years ago. So maybe Veidt is trying to return to Earth to exact revenge? And maybe he is petitioning Dr. Manhattan to help him do it?
Why Dr. Manhattan would care is another question, and here's some speculation: What if Dr. Manhattan created the bubble world on Europa, but intended it only for the life he said he wanted to create at the very end of the comic books, and then Lady Trieu somehow found out about it and somehow figured out how to teleport Veidt into the bubble and then sold him on the idea of building a utopian society there in retirement? It does make some sense that Dr. Manhattan and Adrian Veidt might have a common enemy. Lady Trieu is Vietnamese, and Dr. Manhattan did win the Vietnam War for America; perhaps her parents were killed by Dr. Manhattan, or maybe she herself was injured somehow by him. It's possible that Lady Trieu is playing a long revenge con against Dr. Manhattan.
It's also interesting that Veidt's teleportation technology, which Trieu would now own, is being used by the Seventh Kavalry, and that the Trieu logo can be seen in the back of the Seventh Kavalry truck that Looking Glass followed. Is she in cahoots with them secretly? This article in the Peteypedia could hint that they may want to be teleported off-world to create their own utopian society. Maybe she offered them the ability to go to the same place where Veidt is in exchange for... something? Will did say "tick tock" at the end of Episode 4, which could indicate that he also knows about the conspiracy, and his involvement would make it likely that Trieu is merely using them for some purpose and will double-cross them in the end.
My best guess is that Veidt's prison is one of Lady Trieu's design, possibly piggy-backing off a bubble world originally created by Dr. Manhattan. He could have created two people to inhabit the world, a la Adam and Eve (only these two were made in the likenesses of Dr. Manhattan's human form, Jon Osterman, and his pre-accident flame Janey Slater), and the many versions of them that we see are copies of the originals. Lady Trieu could have outfitted the world with clone technology and other gizmos and sent Veidt there, with the promise that she'd continue his work on Earth. But it was all a ploy.
Now, sequestered away in her compound filled with her own groundbreaking technology plus a life-like statue of none other than Adrian Veidt in his Ozymandias costume (and Adrian Veidt at more or less the age he is now, interestingly enough), Lady Trieu does seem to be running things in Tulsa secretly from behind the curtain.