No. Only the host's save matters — the host loads the edited file and the lobby follows it.
Is this cheating, and can it break my save?
No. The only thing that changes is how long a turn lasts — the game state itself is untouched, so nobody gains any advantage from it. Everything else in the file stays byte-for-byte identical, including the file size, and an edit this small should not damage a save. Even so, keep a copy of the original somewhere until you have confirmed the edited one loads, just to be safe.
What if I pick the wrong game speed?
The editor tells you that the speed does not match instead of writing anything, so a wrong choice cannot corrupt the file.
Can I use it on single-player saves?
The turn timer only exists in multiplayer games, so there is nothing to change in a single-player save.