How much does the sky change in a few thousand years?
The "fixed stars" are not actually fixed, the earth's tilt changes over time etc., but all that happens slowly on human timescales.
Imagine a Babylonian astronomer (or astrologist?) teleported to the present, e.g. to present-day Iraq. Would she be able to tell that the sky is different? If so, what about a Maya astronomer? In other words, how long does it take until the changes in the sky are perceptible to the human eye?
Here's part of the sky in the year 1
It is part of the sky you may know well, Orion and the dogs. I've marked the current positions of Sirius, Procyon and Betelgeuse, with green markers so you can see how their positions have changed over 2000 years. It's not a lot.
The first thing that the Babylonian astronomer might notice is that there is a pole star, one that is fixed. 2000 years ago, Polaris was more than 10 degrees from the True North. Now it is less than 1 degree. It isn't immediately obvious from a casual glance at the sky, but it would be noticeable and surprising to Babylonian astronomer.
If they started looking at the position of stars at a particular time and a particular date, they would find that things are in the "wrong" position. Of course "on a particular date" assumes a calendar, and the Babylonian calendar was not the Gregorian calendar. With more measurement they would find that the point of Aries is now in Pisces. Again, its not something you would notice at a glance, but it is within the ability of a Babylonian astronomer.
Now the stars have actually moved since year 1. But it would not be casually noticeable. It would be within the ability of our Babylonian astronomer to measure, if we asked them to. The relative positions of Sirius and Betelgeuse have changed. The Babylonians could measure angles in the sky precisely enough. But we don't know if they did, and if they did we don't know if they wrote them down.
So our Babylonian would not look up and immediately say "something's wrong". But if we 1. Prepared him by telling them to take careful measurements of Polaris, Sirius and so on. and 2. Gave them some prompts on what to look for in terms of proper motion and precession, then it would be possible for them to observe some changes.
If you don't know if they measured precisely or wrote measurements down, how do you know they *could* make precise measurements if asked? This answer could benefit from some citation (or at least explanation) about the abilities of a Babylonian astronomer.
So... you *do* know that they measured them precisely? Then why say you don't know if they did, if you *do* know?
Measuring accurately the positions over time of the "wandering stars" (planets) is only to be expected. For ancient astronomers to expend equal effort to repeatedly measure **over time** the positions of the "fixed stars" relative to each other would actually be somewhat of a surprise.
FWIW, the Babylonian / Chaldean astronomers used ecliptic longitude & latitude for recording the positions of the planets & the Moon. Some later Babylonians *may* have deduced that the First Point of Aries is moving relative to the stars; after all Hipparchus used Babylonian data (in conjunction with data from other sources) to deduce the precession of the equinoxes.
The Babylonians left daily records spanning over 700 years. See Babylonian astronomical diaries for details, including links to translations. It's not exactly riveting reading material, but I'm very glad we still have those records, and that people have made the effort to translate them.