This article would seem to suggest that use of radio telemetry started being used sometime during the mid-fifties.
Then you reduce these rolls of oscillograph readouts and learn to read the data, you will know more about what happened during a flight test than the pilot, the engineer and the designer. These rolls of paper are like novels. It is up to you to get the meaning, then sense the plot and determine whether flight objectives were satisfied.
The quote above is from chapter 5 of "Failure is not an option" by Gene Kranz, describing his first introduction to flight testing in 1954 when he started working for McDonnell Aircraft. As the quote would indicate, data processing happened after the flight was over and that there was no kind of control room monitoring the progress of the flight. An exception was for incidents such as one also described by Kranz, where a B-52 had lowered a missile from the bomb bay in order to fire it. The missile would neither fire or retract, but as the bomber had enough fuel the engineers had enough time to discuss various options.
I suspect that the first form for flight operations support would have been the chaseplane, although this is outside what you are asking.
The next closest thing I've heard about is Mercury Control, which was used to monitor and support flights during the Mercury Program.