The relation of the speed of light $c$ to electrodynamics was known before Maxwell.

In 1846, Weber derived his force law between point charges:<sup>1</sup>

$$F=\frac{ee'}{r^2}\left[1-\frac{1}{2c^2}\left(\frac{dr}{dt}\right)^2+\frac{1}{c^2}r\frac{d^2r}{dt^2}\right]$$

from [Ampère's force law][1]<sup>2</sup> (not to be confused with one of Maxwell's equations, the [Ampère circuital law][1]) between current elements:

$$d^2\vec{F_{21}^A} = - \frac{\mu _0 }{4\pi }I_1 I_2 \frac{\hat {r}_{12} }{r_{12}^2 }\left[2(d\vec {\ell }_1 \cdot d\vec {\ell }_2) - 3({\hat {r}_{12} \cdot d\vec {\ell }_1 })({\hat {r}_{12} \cdot d\vec {\ell }_2 })\right] = - d^2\vec{F_{12}^A}.$$

In the form Weber wrote his law, it involved a constant $a$ that is related to the speed of light $c$ by a factor of $\sqrt{2}$.

In 1856, Kohlrausch & Weber experimentally determined the constant.<sup>3</sup>

In 1857, Kirchoff explicitly tied this constant to the propagation of electricity in a wire:<sup>4</sup>

>The velocity of propagation of an electric wave is here equal to $\frac{c}{\sqrt{2}}$; it is therefore independent of the cross-section of the wire, of its conductivity, and, finally, of the electric density; […] it is thus very near the speed of light in empty space.

See [Assis][2]'s [*Weber's Electrodynamics*][3] and ch. 8 of Duhem's [*The Electric Theories of J. Clerk Maxwell: A Historical and Critical Study*][4].

<hr>
References

   1. W. Weber, [*Elektrodynamische Maassbestimmungen*][5] [[*Determinations of Electrodynamic Measure*][6]], Leipzig, 1846: p. 142 ff. of Weber's [*Werke* vol. 3][5] or p. 81 (§18 or PDF p. 82) ff. of [this translation][6] (also [here][7]). In 1848, Weber wrote a shorter paper, "[On the Measurement of Electro-dynamic Forces][8]" (also [here][9]); see spec. pp. 32-43 for the derivation of his law.

   2. First published English translation:

      - Assis, André Koch Torres, J. P. M. C. Chaib, André-Marie Ampère. 2015. [*Ampère's electrodynamics: analysis of the meaning and evolution of Ampère's force between current elements, together with a complete translation of his masterpiece: Theory of electrodynamic phenomena, uniquely deduced from experience*][10]. (also [here][11])

   3. R. Kohlrausch and W. Weber, *Elektrodynamische Maassbestimmungen, insbesondere Zurückführung der Stromintensitäts-Messungen auf mechanische Maass*, Leipzig, 1856. [English translation: [Weber and Kohlrauch (2003)][12]].

   4. G. Kirchhoff, [*Ueber die Bewegung der Elektricität in Drähten*][13] [[On the motion of electricity in wires][14]] (*Poggendorff’s Annalen*), Bd., 1857. [English translation: [Kirchhoff (1857a)][14]].


  [1]: https://hsm.stackexchange.com/q/5059/232
  [2]: http://www.ifi.unicamp.br/~assis/books.htm
  [3]: https://isidore.co/calibre/browse/book/4302
  [4]: https://isidore.co/calibre/browse/book/4976
  [5]: https://isidore.co/calibre/browse/book/4654
  [6]: https://isidore.co/calibre/browse/book/4310
  [7]: https://www.21stcenturysciencetech.com/Articles%202007/Weber_1846.pdf
  [8]: https://isidore.co/calibre/browse/book/4861
  [9]: http://www.sizes.com/library/classics/Weber1.pdf
  [10]: https://isidore.co/calibre/browse/book/5464
  [11]: http://www.ifi.unicamp.br/~assis/Amperes-Electrodynamics.pdf
  [12]: http://www.ifi.unicamp.br/~assis/Weber-Kohlrausch(2003).pdf
  [13]: https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/6TG7ABBC/Kirchhoff%20-%201857%20-%20Ueber%20die%20Bewegung%20der%20Elektricit%C3%A4t%20in%20Dr%C3%A4hten.pdf
  [14]: https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/335PCK2X/Kirchhoff%20-%201857%20-%20LIV.%20On%20the%20motion%20of%20electricity%20in%20wires.pdf