For a long time it was not only believed but even ascertained that electric signals moved not just as fast but faster than light, even "instantaneously". The original experiments involving electrostatic discharges of the Leyden jar were made even before wires were introduced. According to Fahie's History of Electric Telegraphy, one of the early experimenters, Winkler "in 1744, ascertained that the rapidity of an electric discharge was exceedingly great and comparable with the speed of lightning". He used a battery of three jars connected by an insulated wire, laid along the bank of the river Pleisse, whose waters closed the circuit. Lemonnier made another demonstration:
"In April 1746, in the court of the Carthusians, he so laid out two parallel wires of 5700 feet each, that all four ends were close together. Between one pair he placed a jar, and grasped the other extremities himself; then on causing the circuit to be completed, he could not distinguish any interval (so short was it) between the spark at the jar, and the shock through his arms."
In 1746 Nollet performed experiments on the propagation speed with 200 monks hand in hand, forming a circle of about 1.6 km and concluded that the speed of electricity was very high but finite, see Guarnieri, The Rise of Light. But 1747 grand scale systematic experiments conducted by a committee of the Royal Society under Watson including Folkes, Cavendish and Bevis, among others, led to a different conclusion:
"...On the 14th August at Shooter's Hill, an experiment was made "to try whether the electric shock was perceptible at twice the distance to which it had yet been carried, in ground perfectly dry, and where no water was near; and also to distinguish if possible its velocity as compared with that of sound." The circuit consisted of two miles of wire, and two miles of perfectly dry ground, but one shower of rain having fallen in the previous five weeks. The wire from the inner coating of the jar was 6732 feet long, and was supported all the way upon baked sticks, and that which communicated with the outer coating was similarly insulated, and was 3.68 feet long.
The observers placed at the ends of these wires, two miles apart, were provided with stop watches with which to note the moment that they felt the shock. The result of a series of careful observations was that "as far as could be distinguished the time in which the electric matter performed its circuit might have been instantaneous"". [quoted from Fahie]
It should be noted that at the time gravity was also believed to act instantaneously, to Newton's chagrin, and certainly faster than light. In Celestial Mechanics (1799) Laplace introduced velocity dependence into the gravity law, and showed that the planets would quickly fly off of their orbits unless gravity was millions of times faster than light, see What 19th century developments contributed to the General theory of Relativity? As late as 1837, the instantaneous propagation of signals in the wire was still asserted in Alexander's telegraph proposal, based on Ampere's and Ritchie's ideas, and published in multiple Edinburgh and London journals:
"It has been found by experiments made with a view to ascertaining the velocity of electricity, that it is transmitted instantaneously, by means of a common iron wire, a distance of eight miles; and electricians of the first eminence have declared their opinion that, judging from all scientific experience, the electric or galvanic influence would be almost instantaneously transmitted from one end to the other of a metallic conductor, such as ordinary copper wire of moderate thickness, of some hundred miles in length." [quoted from Fahie]
According to Guarnieri, in 1854 Thomson (later Lord Kelvin), while laying the transatlantic telegraph cables, derived the first version of the "telegrapher's equation" (second-order PDE) for the electromagnetic propagation in a telegraph line. But he ignored inductance and treated telegraphy as a diffusion rather than wave propagation. Hence the signal speed was inverse proportional to the square length of the cable. In 1857 Kirchhoff (1824–1887) derived a wave equation for the current and charge in a long line, taking into account self-induction effects, and computed that the speed was equal to the speed of light. The modern derivation of the telegrapher's equation from Kirchhoff’s circuit laws was first given by Heaviside in 1876. In 1886 he introduced the term impedance, and in 1887 developed the modern model of a “distortionless” transmission line, see historical note in McDonald's Distortionless Transmission Line.
Back in 1855 Weber and Kohlrausch noted that "the ratio of the absolute electromagnetic unit of charge to the absolute electrostatic unit of charge", in modern notation $\frac1{\sqrt{μ_0ε_0}}$, where $μ_0,ε_0$ are the magnetic permeability and electric permittivity of the vacuum, respectively, had the units of velocity, and determined experimentally that it was remarkably close to the speed of light. Towards the end of 1861 Maxwell derived the general formula $v=\frac1{\sqrt{με}}$ for the electromagnetic propagation speed in a medium in part III of his paper On Physical Lines of Force, and suggested, in particular, that light was a form of electromagnetic radiation, see History of Maxwell's equations. It also resolved the issue of the speed of electromagnetic propagation in a medium.
And, as it turned out, Laplace's estimates were based on a wrong sort of velocity dependence (in modern terms, non-Lorentz invariant). Electromagnetic theories of gravity, implying its propagation at the speed of light, were offered at the end of 19th century, notably by Lorentz himself.