Another way would probably be to have one of both universes be fiction inside the other..and then something happens. For example: There is nothing to say that at least the old Dr. Who series did not exist in Trek universe (the new show would not have existed since its creation would fall right into WWIII if I´m not mistaken).
No, Trek's WWIII is still several decades in our future.
We do know that Star Trek
exists as fiction in the canonical Doctor Who
universe. "The Empty Child" had Rose referring to the Doctor as "Spock," and there have been various other Trek allusions in the following seasons of DW and The Sarah Jane Adventures
. (Which makes the current IDW Trek/Who crossover miniseries problematical with respect to DW continuity -- but then, DW has never had much continuity or logic anyway.)
There's never been a canonical reference to DW as fiction within the Trek universe, but the novel My Enemy, My Ally
featured Uhura and Lt. Freeman converting Tom Baker episodes to holographic format (although the clip "shown" in the book is not a scene that actually occurs in any real DW episode).