I've started a bunch of games (20 +) post-Patch, but haven't played many past the Classical Era (which is my usual pattern in Civ VI) so here are my observations from the Early Game only:
Pre-Patch, the AI seemed to go for, in this order:
Stonehenge - never had a game when it wasn't built by about Turn 25 at the latest, pre-GS
Hanging Gardens
Temple of Artemis
- After that it gets very random, depending on the type of map, number of opponents, amount of coast, etc. However, the least built always seemed to be the Mausoleum. I'm a Wonder Hog, and I don't think even I have built it more than 2 - 3 times in 2100 + hours of playing since Vanilla, and in the majority of games I've seen it built as late as 2 - 3 Eras after it becomes available.
Post-GS/Patch, the earliest Wonder always seems to be the Great Bath, closely followed by Stonehenge and Artemis, but it is much less certain: I've had two games in which neither Great Bath nor Stonehenge were built until after Turn 50, which NEVER happened in any game Pre-Patch or Pre-GS.
Rule of Thumb for me now, though, is that if I can't get Great Bath by Turn 25, don't bother, because with the exception of those two 'outlier' games noted above, Great Bath is always completed between Turns 24 and 29 by someone (very rarely me!).
The early 'naval' Wonders: Great Lighthouse, Colossus, are the most random. On a 'naval' map like Small Continents, Archapelago or the new Seven Seas, or with a bunch of 'naval oriented' Civs like Carthage, Indonesia, and England, they go fast. On a Pangaea or a random selection of Civs that does not include anyone with a Naval UU, they may not be built until the late Medieval Era or later.
Surprisingly, even the 'Domination-Oriented' Civs like Mongolia and Zulu do not seem to prioritize the Terracotta Army: I can count on the fingers of one hand the games in which I did not manage to build it, even when I was playing a peaceful game in which I did not in any way make it a priority. On the other hand, Apadama is almost always completed very soon after the required Tech level is reached and I have seen games in which 4 out of 9 AI opponents were all trying to build it at the same time. I don't think I've seen that level of Interest in a single Wonder other than Great Bath and, in per-GS games, Stonehenge.