According to the Memory Alpha page, this occurred sometime in the 2340s.
However, more specifically, according to Star Trek Chronology, the following is said:
2347: Starfleet Command orders a change in the design of the Starfleet emblem. The new simplified design features an elliptical background field and a slightly redrawn arrowhead
The book does note that this is conjecture and it does not note specifically why they chose 2347 as the year, however we can be certain it occurred between 2344 and 2349 because we see in Yesterday's Enterprise, the crew of the Enterprise-C is using the TMP-style insignia (the crew on board was from 2344), whilst the insignia on Jack Crusher's uniform in the hologram he created for Wesley* in 2349 (seen in Family) is using the TNG-style insignia.
*As an interested aside, it is noted in the book that this was an accident - when filming the hologram scene, Producer Ronald Moore is supposed to have noticed that Jack wasn't wearing an insignia and when he pointed this out the only insignia they could find was a TNG one