No, not the doctors of St Elsewhere. A Doctor Named Who. Or, actually, not named Who. We don't know his name. His show is Doctor Who and it's a piece of science fiction legend.
Sitting here watching Doctor Who's "The Rise of the Cybermen", I remembered this post by eljuno - which brings in a few new shows into the Tommyverse.
The nexus is another British legend, Sci Fi sitcom Red Dwarf. In the episode "Psirens", the crew of the Red Dwarf come across a space ship graveyard - which includes a Klingon Bird of Prey from the Star Trek universe, a Weyland-Utani ship from the Alien series of films (Weyland-Utani also part of Angel and Firefly canon), as well as an Eagle ship from the TV series Space: 1999.
Given the predeliction for parallel universes in science fiction series - particularly in Trek but also in Dwarf and Who (see "The Rise of the Cybermen"), this may be Tommy Westphall Parallel Dimension time, but they are still joined, uh, at the ship...
Oh, Doctor Who. The Doctor travels through Time and Space in a ship called a TARDIS, which looks like an old British Police Box due to a broken chameleon circuit. (Ooh, look who's up on his Who lore!) The TARDIS appears in the hangar bay of the Red Dwarf - it can be seen during the launch of Starbug in the episode "Thanks for the Memory".
The Star Trek universe is already an established part of the Westphall-verse: linked to Angel and The John Larroquette Show through the fictional Yoyodyne company. (Yes, we're aware of its connection to/origin in the Buckaroo Banzai movie.)
So, through Trek to Red Dwarf we also get Space: 1999 and Doctor Who. Now I don't know if Space: 1999 already connects to other shows, but Who certainly does...
K9 and Company was a one-off episode broadcast as a Christmas Special in 1981 - featuring The Doctor's old companions K9 - the robot dog - and Sarah Jane Smith. Produced as a pilot for a proposed TV series, that option was never picked up, but a one-off aired pilot does count in the Westphall-verse. (K9 and Sarah Jane returned to the Whoverse this year in the episode "School Reunion".)
Torchwood will begin airing in October. The history of the covert Torchwood Institute was seeded in the second season (or the 28th, depending on how you count them) of Doctor Who - and the main character, Captain Jack Harkness - a bisexual space adventurer and rogue, was introduced in season one (or 27).
Sarah Jane Investigates is a pilot for a proposed series starring Elizabeth Sladen as Sarah Jane Smith. The pilot is tentatively scheduled to air on January 1, 2007. A decision about a move to series will presumably come after that point. But, as with K9 and Company (and W*A*L*T*E*R and The Art of Nick), an aired pilot counts.
NOTE: The Westphall Universe only includes television programs, not films or books or comics or animated series. With Doctor Who's nearly forty years of history, I'm betting a lot of connections will surface to other British programs. There are a few televised Who stories that are not considered canon, however - particularly "Dimensions in Time" (which would have brought in Eastenders) and "The Curse of the Fatal Death" (which was a Comic Relief special).
With the inclusion of Dwarf, Who, Torchwood, K9 and Sarah Jane, the British arm of the Westphall-verse increases dramatically from its previous single entrant - the original version of The Office. Other than these, the only non-US series in Tommy brain come from the Canadian Degrassi mini-verse.
In summary: Trek brings in Red Dwarf, which brings in Space: 1999 and Doctor Who, which brings in K9 and Company, Torchwood and the upcoming Sarah Jane Investigates.