I don't know about oblivion so much, but the situation you have created is not how the mod was intended to be used.
The Traveler mod procedure to move sims to a different world (more or less) permanently is to vacation them there, have them purchase a vacation home (change an unoccupied one in Edit Town to Player Ownable if you need to or else there might be no vacation homes to purchase), move into it fully so they are no longer staying in the free housing provided for by the mod, invoke Traveler's Change Hometown on them, save (as), quit to desktop or Origin, and reload.
http://nraas.wikispaces.com/Traveler%20FAQ
(see the Switching Homeworlds sections towards the end)
If I know in advance that this is how I want things to go, I might have the vacationing sims "stay" on an empty residential 10x10 I would have set aside for this purpose beforehand, and have them purchase and move into the home I really want them to own almost right away after arrival. Then as above.
The procedure is not to travel your sims to a new (to them) world and abandon them there by switching households and ignoring their vacation clock as it winds down. Nowhere in the documentation does it suggest doing that. What I believe happens then is that the connection between the instance of MH you are playing and the AP homeworld becomes broken and there is no way to return to it from where you are now, besides reverting to a much prior save and re-doing the ending of the original vacation another way. The game will be stuck somewhere between vacation and regular homeworld play mode and you are in unsupported territory by then.
Edit: Actually, re-reading this now, it is okay to switch households briefly, not permanently, while in vacation mode. The Traveler mod should react by changing its Pause Travel setting to True and its Treat as Vacation setting to False for you since of course the other household presumably lives there already. But the step missed was that at some point relatively soon you must switch back to the household that was on vacation, unpause Travel and change Treat as Vacation back to True if this doesn't happen automatically, and then either proceed with the remaining tasks to switch homeworlds on them (buy vacation home, move into it, etc.) or else end the vacation by sending them back home.
There are reasons we list these steps out as precisely and carefully as we have. The game is fragile enough as it is without adding more sources of "what happened to my homeworld?" to it by undertaking creative but less than supported actions on what should be connected worlds. I'm sorry if it sounds like I am scolding here, I really don't mean to be, but I just cannot understand why so many players ignore the documentation that comes with these mods and do things so totally differently. You certainly aren't the only one who has played their game into a corner like this over the years! :/
I also don't know why you have two Appaloosas. I mean, I understand how that happened from the way you described things, but the game does not understand. As far as the game is concerned, you are either playing an ongoing save with the Pets EP in place and using the stock AP world designed by EA and installed by the Pets EP, or you aren't. As near as I can tell, you cannot do both. Maybe I misunderstood where the two .world files are coming from. If Pets is not in play at all and hasn't been anywhere along the line here, do you have two entries in your InstalledWorlds folder in the user game folder in Documents?
Having said all of this now, I will also point out that besides all the procedures not exactly being followed, some custom worlds just do not work well with Traveler because they are not set with a unique ID number in their creation stage. So travel away and back again to/from these worlds just does not work and new instances of them are always created upon re-entry. The latest versions of Traveler (v87 or the current beta testing v88 version) help cut through some of that, but it's not always going to be successful. It's best to test custom worlds off to the side with some test regular vacations and moves back and forth first, before using them on a "real" game to be so connected, so that you would know in advance whether the worlds even have a chance to work out well in an ongoing rotational back and forth type of game. Given the other variables above though, it is impossible to say if the custom AP you are using is contributing to the problem or not because of this exact issue.