Uranus was discovered in 1781, Pluto in 1930. How long did it take, or did they know pretty much immediately? I’d think that for several years at least they were just tiny specks in the nighttime sky.
I was going to say something about asking what can be obtained from Princess Google. But I see it’s not as easy as anticipated. Took me more than five minutes to find an answer (an eternity for a Google search!). 
Two satellites of Uranus were discovered in 1787, and moons usually orbit in the equatorial plane of their planet. I would therefore suspect that early astronomers could have deduced that Uranus had a large axial tilt once these satellites were discovered. Similarly, Charon (Pluto’s largest moon) was discovered in 1978, so I would expect that astronomers would have concluded that Pluto had a strong axial tilt by that time if not before.
That said, I have no idea whether they actually did deduce this; or (particularly in the case of Pluto) whether there was other evidence of a strong axial tilt that preceded these discoveries.
ETA: Here’s Herschel’s 1787 paper on the discovery of planets orbiting Uranus (or “the Georgian planet”, as he proposed it be called.) Towards the end of the paper, he notes that “their orbits make a considerable angle with the ecliptic.”
William Herschel is credited with Uranus’ discovery in 1781 when he observed a distinct disk through his 6" telescope. That’s not big enough to track the swirls in the atmosphere, but it wouldn’t be long after for the 18" or 24" telescopes to see this. Unfortunately my google-fu is weak today and I can’t seem to find the details of this particular discovery.
The details on Uranus are found at the link I posted.
What is the reason for that? Does the satellite’s orbit influence the direction of rotation, or vice versa?
It’s where the moons formed and nothing has moved them away since then.
When large planets form, they will have an extensive disc around them, much like the Sun had when it formed. Some of that material will eventually spiral into the planet, but the rest will collect into satellites.
Not all satellites formed this way. Mars’s moons are thought to be captured asteroids, while most other small objects (Earth, various asteroids and TNOs) have satellites that were probably the result of collisions.