Are you asking about the concept of speed in general or the equational relationship? I can highlight a few developments.
The concept of speed was debated by the Greeks. For instance, Zeno of Elea presented two famous paradoxes regarding speed, and Aristotle addresses them in his Physics. From Wiki:
Achilles and the Tortise
Achilles is in a footrace with the tortoise. Achilles allows the
tortoise a head start of 100 meters, for example. Suppose that each
racer starts running at some constant speed, one faster than the
other. After some finite time, Achilles will have run 100 meters,
bringing him to the tortoise's starting point. During this time, the
tortoise has run a much shorter distance, say 2 meters. It will then
take Achilles some further time to run that distance, by which time
the tortoise will have advanced farther; and then more time still to
reach this third point, while the tortoise moves ahead. Thus, whenever
Achilles arrives somewhere the tortoise has been, he still has some
distance to go before he can even reach the tortoise. As Aristotle
noted, this argument is similar to the Dichotomy. It lacks, however,
the apparent conclusion of motionlessness.
Zeno states that for motion to occur, an object must change the
position which it occupies. He gives an example of an arrow in flight.
He states that at any one (duration-less) instant of time, the arrow
is neither moving to where it is, nor to where it is not. It
cannot move to where it is not, because no time elapses for it to move
there; it cannot move to where it is, because it is already there. In
other words, at every instant of time there is no motion occurring. If
everything is motionless at every instant, and time is entirely
composed of instants, then motion is impossible.
Whereas the [Achilles and the Tortise paradox] divide[s] space, this paradox starts by dividing time—and not into segments, but into points.
These are interesting paradoxes to consider, especially in the context of calculus.
According to science and math historian Marshal Clagett, Aristotle "is the first one whose treatise we have to give us rules for comparing the speeds of bodies in terms of the space traversed in given times...we can derive from [four inequalities between space, time, and speed] our general expression V=S/T once we have decided that velocity can be represented as a ratio."
This is probably more what you're looking for.
Also, it's worth noting that Aristotle proposed that the speed at which two identically shaped objects sink or fall is directly proportional to their weights and inversely proportional to the density of the medium through which they move. This was a major point of contention for Galileo as he showed this proposition to be false.
In the Third Day: Uniform Motion of Two New Sciences, Galileo restates the definition of uniform speed: "distances traversed by the moving particle during any equal intervals of time, are themselves equal."
He further adds the condition "the word 'any,' meaning by this, all equal intervals of time; for it may happen that the moving body will traverse equal distances during some equal intervals of time and yet the distances traversed during some small portion of these time-intervals may not be equal, even though the time-intervals be equal."
From there, he states that the definition of uniform speed requires Axioms I-IV (the inequalities mentioned earlier), then uses them to construct ratios of speed, distance, and time in Theorems I-III, and then shows the product (i.e., compound) of these ratios in Theorems IV-VI.
Although Galileo famously presents his version of the mean value theorem for uniform acceleration, the mean value theorem was actually demonstrated by the Oxford Calculators in the 14th century, and was later proved by Nicole Oresme. The theorem is also known as the "Merton Rule": https://en.wikipedia.org/wiki/Mean_speed_theorem
Lastly, I should note that, in all the developments above, the arguments were presented using geometry and not algebra. The equation V=S/T would be objectionable to everyone mentioned above because dividing one magnitude (distance) by a different magnitude (time) to produce a third magnitude (speed) was considered absurd -- it would be analogous to dividing angles by areas or arc lengths by volumes. Instead, in Euclidean geometry, like magnitudes were compared to like magnitudes through ratios. For example, Galileo presents the relationship V1/V2 = (S1/S2)x(T2/T1) where "/" signifies ratios and not algebraic division, and "x" signifies a compounding of ratios and not a multiplication or product.