As far as I understand, the Kaluza-Klein theory, despite its unprecedentedly profound and beautiful character, had a modest following in its early days. I guess that two of the many reasons might be the lack of additional experimental predictions by the theory and the reluctance towards accepting the possibility of additional spatial dimensions. Anyway, the idea of additional spatial dimensions, although in a somewhat different context, was revived by the rise of String Theory in the 1980s and today, we know the Kaluza-Klein theory is mentioned as Kaluza-Klein Miracle at many places (including my GR classes :P). I wonder if the rise of String Theory made the old Kaluza-Klein idea a sensation. Or was Kaluza-Klein theory equally appreciated and loved even before?
The answer depends on the meaning of "appreciated". One could say that it was appreciated from the start, Einstein commented on the idea positively back in 1919, and in 1921 he presented Kaluza's paper “Zum Unit ̈atsproblem der Physik" to the Prussian Academy. The reason Kaluza-Klein theory did not receive the kind of recognition it would have received in 19-th century is that in 1920s the focus was on quantum mechanics, in 1930-40s on quantum electrodynamics, and later on quantum gauge theories, emphasis on quantum. The classical unification programme, to which Kaluza-Klein was the most successful entry after Maxwell, however dear to Einstein's heart, came to be seen as outdated. If there was a hope of unification it head to unify quantized theories, and Kaluza-Klein was not that. Pauli's evolution on the issue is characteristic, he supported Weyl's 1918 conformal unification theory, but by 1920-21 he sounded a different tune:
"I would like to ask Einstein if he agrees that one can only expect a solution to the problem of matter by a modification of our ideas about space (and perhaps also time) and electric fields in the sense of atomism...or should one hold on to the rudiments of the continuum theories? [...] New elements foreign to the continuum must be added to the basic structure of the theory before one can arrive at a satisfactory solution of the problem of matter."
This is pretty much the majority position today. Sure enough, Einstein, given his attitude towards quantum mechanics, was unimpressed. In fact, he intensively studied Kaluza-Klein theory in 1938-1943 in hopes of finding an alternative to quantum mechanics, see van Dongen's Einstein and the Kaluza-Klein particle. That did not work out.
The revival of interest in the theory was due to Witt's 1964 observation that by adding more dimensions one could unify gravity with non-Abelian gauge theories. Of course, the unification was still classical, but the idea took off and was indeed boosted by extra dimensions in the nascent string theory and later supergravity. Chodos's 1984 overview of Kaluza-Klein theories describes the situation in the early 80-s, which one could call their second appreciation:
"In the late seventies, spurred by recent progress in higher-dimensional supergravity theories, and by the fact that the grand-unified scale was only a few orders of magnitude above the Planck length, the Kaluza-Klein idea suddenly acquired widespread notoriety, with the result that by now the diligent researcher who wishes to become acquainted with the field is faced with a reading list of several hundred papers, all written within the last few years."
Here is a paper that features the early history of the Kaluza-Klein theory. It seems that one of its earliest supporters was Pauli. When Victor Weisskopf became Pauli's assistant in 1933, he told him that he would work on anything but the Kaluza-Klein theory, claiming he did not understand it: https://books.google.de/books?id=LdMVAwAAQBAJ&pg=PA281