In terms of trek/Dr. Who cross-overs, for the artists out there:
TOS meets fourth Doctor
Tennant should have been on Enterprise-D (more work with Pat)
Matt Smith with JJ and Pine.
It gets a bit harder after that.
Hartnell on the Orbit Jet.
Second Doctor on Lost in space or the Time Tunnel
Third Doctor On Voyage to the Bottom of the Sea...or UFO
Fifth Doctor in Buck Rogers with Erin Grey
Sixth Doctor on Red Dwarf of course.
Seventh Doctor...hmm NuBSG--Stargate--or, of all things Game of Thrones.
Eighth on Space Above and beyond
Ninth in Alien
If, by some miracle of cross-channel and transAtlantic co-operation, you could have had Trek/DW crossovers take place while productions of each were ongoing, then the crossovers could have involved:
1966: Original Star Trek crew and The First Doctor (very narrow window of opportunity, as Trek debuted in September 1966 and the first Doc regenerated at the end of October 1966)
1967 - 1969: Original Trek crew and the Second Doctor. The Second Doctor's tenure and the last episode of Trek both debuted in June 1969.
For the Third Doctor to have teamed up with the Trek crew, it would have had to be as an appearance in the animated series.
Tom Baker debuted as the Fourth Doctor about 2 months after TAS stopped airing but he was still in the role at the time that the Trek crew reunited to make TMP.
Peter Davison's Fifth Doctor could have teamed up with the red-uniformed Trek crew around the time of TWOK or TSFS.
Colin Baker as the Sixth Doctor could have crossed over with the Trek crew around the time of TVH.
Colin was gone by the time that TNG aired but that Trek crew and Sly McCoy debuted in the same month, September 1987. McCoy could also in theory have appeared in the fifth Trek movie, TFF.
DW isn't on air for the 6th and final big screen outing of the original crew of the Enterprise (TUC, 1991) or the TNG crew's big screen debut in GEN (1994). However, Paul McGann's only onscreen appearance as the Seventh Doctor comes in a year when DS9 and VOY are on air and the TNG crew are on the big screen in FC.
Chris Eccleston's only series as the Ninth Doctor just about overlaps with the final series of ENT. The latter is scrapped too soon for the Tenth Doctor to have crossed over, but the most recent Trek movie aired in the same year as Tennant's last year as Ten, while Matt Smith and Chris Pine etc are still signed up for their respective roles.