Precision distance and time measurement were probably the first technologies where SR and GR effects were important experimentally. Though obviously you don't need SR and GR to measure time or distance, and both of these have a long, long, history.
Distance measuring, and SR, was probably first impacted. According to wiki, Michelson and Morely did the first interferometric measurment of the prototype meter bar in 1892 or 1893, with more measurments by Fabry and Perot in 1906. Now, you don't need special relativity to have the idea to use interferometric techniques to measure distance, but based on the theories of the time, it was thought that if you did use such techniques, measurements would be distorted by an "Ether wind". But when the magnitude of the expected distortion effect was measured, it came up to be zero.
The Michelson-Morley null resut for the Ether wind this was probably the first indication of the need for relativity, which was proposed later in 1905 by Einstein.
You don't need SR or GR to create a single atomic clock - but you do need to account for gravitational time dilation to set up an accurate, global time coordinate system (TAI time).
According to wiki, atomic clocks were invented in 1949. In the late 70's two important decisions were made. In 1977, corrections to the networked atomic time based on altitude were incorporated into the TAI time standard. So, General Relativity became officially recognized as important in defining the coordinate time TAI time on this date by the need to adjust the rates of atomic clock by the altitude above sea level.
Some other important events:
in the 1960's, interferometric techniques replaced the old meter prototypes. THis was the first step in re-defining the speed of light as a constant. Basically the most precise distance measurements were based on interferometry at the time, it was known that thee was no "Ether wind". However, the speed of light was not precisely defined due to the fact that the frequency of the light wasn't known yet.
In the 1970's, advances in frequency measurements resulted in the redefinition of the metre as being the distance light travels in a certan number of seconds. This resolved some ambiguties found to arise in the old system and clearly set up the basis for using light and interferometric measurements as a tool to do distance measurments, it being the best and most accurate way of carrying out this process.
The speed of light at this point became something that went from being measured, to something that was so accurate and reliable that it became the basis of measurement.Wiki sources:
http://en.wikipedia.org/w/index.php?title=International_Atomic_Time&oldid=473549247
http://en.wikipedia.org/w/index.php?title=History_of_the_metre&oldid=470631072