Direct measurement of time dilation probably would have had to wait until the advent of atomic clocks. But the experiments listed in this section are all from the 30's and 40's:
<http://en.wikipedia.org/wiki/Special_relativity#Status>
Plus I think some would have to go into fission reactors and the bomb.
E=mc^2 and all that.
But it looks like precise tests of general relativity did have to wait a while:
<http://en.wikipedia.org/wiki/Tests_of_general_relativity>
Tests of Einstein's general theory of relativity did not provide an experimental foundation for the theory until well after it was introduced in 1915. Physicists accepted the theory because it correctly accounted for the precession of the perihelion of Mercury, a phenomenon which had long baffled astronomers and physicists, and because it unified Newton's law of universal gravitation with special relativity in a conceptually simple way. (Einstein has been famously quoted as saying that if his theory was falsified, then he would have felt "sorry for the dear Lord.") [citation needed] Despite Einstein's proposal of three classical tests, the theory was without strong experimental support until a program of precision tests was started in 1959. This program has systematically tested general relativity in weak gravitational fields and severely limited possible deviations from the theory. Since 1974, Hulse and Taylor have studied stronger gravitational fields in binary pulsars. In these regimes, on typical solar system scales, general relativity has been extremely well tested.
[...] -- Andy