Inspired by the discussion of the moons of Uranus providing a clue to the planet's axis of rotation in this question, I'm wondering when it was realised that the major satellites are typically located in the equatorial plane of their planets. Our own Moon is a counterexample to this trend, so presumably it would have taken a bit of detective work.
Since the question has sit unanswered for months, I'm trying to give an acknowledgedly incomplete answer.
In summary: By 1700 it would have been a reasonable guess. By 1800 it was becoming an observed trend.
It's hard to decide when the consensus was created on the typical orbits of satellites, because there are only a few data points (planets with satellites) but we can know when those few data points where known.
Obviously, the only data point known in antiquity was the Moon, but by that time the whole idea of satellites was far from mainstream.
In 1610 Galileo discovered the four biggest moons of Jupiter and that they orbit Jupiter in roughly the same plane. However, it took until the 1660s for Cassini to discover features in Jupiter and therefore be able to realise that its moons orbited in the equatorial plane.
By 1655 Huygens had discovered Saturn's rings and Saturn's satellite Titan, both of them in the same plane. Between 1671 and 1684, Cassini discovered Tethys, Dione, Rhea and Iapetus all of them in the same plane of the rings except for Iapetus.
The rotation of Saturn is a bit harder to detect, and it wasn't measured until 1794 by Herschel.
There is an additional data point even before being able to see Saturn rotating: both Jupiter and Saturn appear elliptical in the 17th century telescopes. Although a causal relation with rotation wasn't established until Newton, as soon as Jupiter's rotation had been discovered it was known that the planet was rotating around the smallest axis of the ellipsoid.
No more data points were added for a long time, because the moons of Mars and the rotation of Uranus were discovered much later, although the first two moons of Uranus had been discovered by Herschel in 1787.
Additionally, it was known from antiquity that all planets were in similar planes, and when sunspots were observed (by Galileo) and the rotation of the Sun could be measured, the Sun's equatorial plane was close to the planes of all planets orbit. Therefore, when the idea of planets and satellites as "little solar systems" arrived, the idea of the central body rotating in the same plane of the system wouldn't have seemed strange.
Then we can summarise what an informed observer at the end of the 17th century would have known:
- Jupiter's rotation, Jupiter's oblation, and Jupiter's moons were in the same plane.
- Saturn's rings, Saturn's oblation and most of Saturn's moons were in the same plane.
- There were only two known exceptions to the rule of "everything around the same planet in the same plane": Earth's Moon and Iapetus, which aren't in the equatorial plane - although they aren't very far away from it.
With all that at hand, I would say that guessing that most satellites orbit in the equatorial plane of its planet would have been a reasonable guess by 1700.
By 1800 the evidence would have increased with:
- The rotation of Saturn had been actually observed, confirming that most satellites and the rings orbit on the equatorial plane.
- Uranus had two satellites in the same plane.
With that, it seems even more reasonable to suppose that Uranus'satellites were orbiting on its (unknown) equatorial plane.