A new paper explains Voigt's transformations and the beginning of the relativistic revolution:
In 1887 W. Voigt published a paper on the Doppler effect, which marked the birth of the relativistic revolution. In his paper Voigt derived a set of spacetime transformations by demanding covariance to the homogeneous wave equation in inertial frames, and this was an application of the first postulate of special relativity. Voigt assumed in his derivation the invariance of the speed of light in inertial frames, and this is the second postulate of special relativity. He then applied the postulates of special relativity to the wave equation 18 years before Einstein explicitly enunciated these postulates. Voigt’s transformations questioned the Newtonian notion of absolute time for the first time in physics by suggesting that the absolute time should be replaced by the non-absolute time t' = t - vx/c2. Unfortunately, Voigt’s 1887 paper was not appreciated by most physicists of that time.I am not sure that anyone saw the significance of Voigt's paper. A paper last year argued:
The Lorentz Transformation, which is considered as constitutive for the Special Relativity Theory, was invented by Voigt in 1887, adopted by Lorentz in 1904, and baptized by Poincaré in 1906. Einstein probably picked it up from Voigt directly.Einstein did not cite Voigt, but did not cite anyone else either.
Voigt corresponded with Lorentz, but did not send the 1887 paper until 1908, with Lorentz agreeing to credit him after that. From this I deduce that Voigt himself did not realize how his paper related to relativity, and it had no influence on Lorentz or Poincare.
Wikipedia has a good broad overview of the History of Lorentz transformations.
Voigt should certainly be credited for early publication of some crucial ideas about Lorentz transformations. I tend to credit Lorentz and Poincare because they had all the relativity formulas but also because they had a big-picture theory. They clearly understood and explained how relativity followed from Maxwell's equations and the Michelson-Morley and other experiments, and they had the really big ideas -- FitzGerald contraction, local time, covariance, non-Euclidean geometry, etc.