This was noticed when observations of eclipses of Jupiter satellites deviated from
prediction. Before that there could be only speculations (and these speculations existed from antiquity). Jupiter satellites
gave the first hard evidence.
From the very beginning, Jupiter satellites were proposed by Galileo as a natural clock for determination of longitude. For that reason, very careful observations were made, and finally the disagreement with Kepler laws was found.
This happened in 17th century.
Romer and Huygens conjectured that the reason is the finite speed of light and measured it (assuming that Jupiter satellites do obey Kepler laws). They obtained
a number which was not very precise but of the right order of magnitude.
Since then observations were made more and more precise.
By the way, Jupiter satellites gave the most precise way of synchronizing clocks
at different locations (=measuring of longitude) until the invention of telegraph.
But the method could only be applied on land, not on a ship.
That other electromagnetic oscillations are spread with the same speed as light was discovered by Maxwell.