The phrase "a time when everybody" is correct and usable in written English.
It can be used to refer to a specific period in the past or a hypothetical situation where everyone is involved or affected.
Example: "I remember a time when everybody came together to support the community during the crisis."
Alternatives: "an era when everyone" or "a moment when all".
"There suddenly was a time when everybody seemed to take me terribly seriously.
This is a time when everybody in this administration should be doing everything in their power to support you.
Webb's generation of rock stars, as Lisa Hubbard points out, lived in a time when everybody was less mobile and not plagued by ringing cellphones.
Even Murdoch is accorded a sneaking gratitude for his willingness to make heavy investments in the newspaper business, at a time when everybody else seems to be disinvesting.
As Jack Bass, a Southern newspaperman turned historian, observed when we talked the other day, it was a time when everybody watched the three network news programs.
South Korea is a notably egalitarian nation, in large part, perhaps, because its citizens remember a time when everybody was equal — and poor.
But for them it is an exciting time, when everybody has a device and display readily available, which can be put to good use in the future.
The early days of "free jazz" were a "fantastic time when everybody arrived in New York City: Coltrane, Pharoah Sanders, Cecil Taylor, Cherry.
"The Sound of Music" pulled people back, if only briefly, to the age of pre-sundering — to a prelapsarian time when everybody watching the Oscars on TV could tell you exactly what a resourceful nun could do with unwanted drapes.
This could be dinner, or a calm time when everybody is hanging out in the living room.
"There was certainly a long period of time when everybody talked the talk, but nobody walked the walk".
