Don't think that picture shown above is just some photoshop joke! In the Toobworld Dynamic, 'Star Trek' and 'Doctor Who' share the same TV Universe. No matter where in the galaxy the Enterprise could be, the TARDIS could easily reach it AND breach its defenses when it materialized. (I can't remember the source, nor the specifics as to which episode, but at one point the TARDIS was seen tucked away in the docking bay.)
And as for getting hold of a Starfleet uniform and taking the command chair (probably during third shift?), Benjamin Sisko pulled it off and that was without the crutch of the psychic paper. So the Doctor could have done so easily with that handy little device.
I'm fairly certain he would have gotten away with the deception......
Although not an intentional link, we can claim that the two shows share the same dimension because the Doctor visited a planet called Vulcan. From the Doctor's point of view, it was an uninhabited world until human mining colonists arrived there. And there were Daleks in stasis buried at a crash site.
To offset the contradictions with the homeworld of the Vulcan people as seen on 'Star Trek', there must be more than one planet in that particular solar system of Mutter's Spiral. Vulcan I would be Spock's planet of origin, while Vulcan II would be a moon farther out in the rotation around their sun.
Vulcan may not have been the only planet from 'Star Trek' visited by the Doctor. We here at Toobworld Central like to take screen captures from the movies (but not from other TV shows) and apply them to Toobworld situations involving the same actors seen in the pictures.
I know that makes me a fanfic enabler. So sue me.
Here's a picture of William Hartnell in some Brit noir. (I'll try to find out the name of the movie.) That's how it has to play in the Cineverse, but in Toobworld, why can't this be a picture of the First Incarnation of the Doctor as he tried to blend in with the citizens of Sigma Iotia II, where the society had been infected by knowledge of the mob wars in Chicago of the 1920's? (I bet there was a chapter in that book about Chicago's connection to Nucky Thompson's 'Boardwalk Empire' in Atlantic City......)