In the late sixties, civic unrest in the Wizarding community took the form of pure-blood riots in response to Squib Rights marches (Pm, JKR-W3).
It is likely that Idris Oakby, the famous campaigner for squib rights, would have been involved in some way. She would have been nearly 100 years old at the time, but she lived for another two decades after the marches in the 1960s (FW, JKR).
The date would be either 1968 or 1969, since Rowling refers to the pure-blood riots as having taken place in the "late sixties," and Jenkins took office in 1968. The date assigned, 1968, is chosen to coincide both with Jenkins taking office and the worst of the race riots in the United States.