The LEGO Movie (franchise)
The LEGO Movie is a multimedia franchise which began with the computer-animated brickfilm of the same name, and went on to have various spinoffs (of sorts), such as The LEGO Batman Movie, as well as a sequel to the original film, titled The LEGO Movie 2: The Second Part. In addition to these, it also has various adaptations into other mediums, such as video games and theme park attractions, as well as several pieces of merchandise, primarily LEGO sets.
The original film stars the main character of Emmet Brickowski, a construction worker living in Bricksburg who winds up with the Piece of Resistance attached to his back. Through the main story, he meets various characters, including allies such as Wyldstyle, Vitruvius, and Unikitty, as well as villains, such as Bad Cop and Lord Business. The series is ultimately set within a diagetic LEGO world, actually in the basement of "the Man Upstairs". His child, Finn, is the one playing with the LEGO characters, although the LEGO characters seem to sometimes possess genuine autonomy, especially in The Second Part.
Crossovers
The franchise has had several crossovers with the Doctor Who universe.
The main crossover between the two franchises, which many of its other crossovers are promotion for, is the video game LEGO Dimensions. The game allows the player to scan in various characters from different franchises to play as, including characters such as both the Twelfth Doctor and Wyldstyle. Wyldstyle, in particular, is actually one of the three main protagonists in the game, with the character having several interactions with the Doctor, and various other elements from the Doctor Who series, such as the Dalek Emperor.
Through this game, naturally, it is possible for Doctor Who characters to use vehicle and objects from The LEGO Movie franchise, and explore its adventure worlds, and vice versa.
Shortly before the game's release, a promotional video was released entitled Endless Awesome. This video featured characters such as the Twelfth Doctor appearing alongside the likes of Wyldstyle and Emmet Brickowski. Most notably, it featured a short scene where Unikitty jumps out of the Doctor's TARDIS, much to his surprise.
Later, after the game's release, various other promo videos were also released, containing original animations depicting both franchises as coexisting. First, the webcast Doctor, Doctor, Doctor primarily featured 3 doctors facing off: the Twelfth Doctor, Peter Venkman (Ghostbusters), and Doc Brown (Back to the Future). In addition to these characters, Good Cop/Bad Cop from The LEGO Movie can be seen in a brief cameo, seemingly just happening to be present on the Brick Boulevard at the time.
Next, the video New Adventures Await!, mainly promoting the upcoming additions to the game such as Adventure Time and Sonic the Hedgehog, featured as its main characters the same trio from LEGO Dimensions: Batman (DC Multiverse), Gandalf (The Lord of the Rings), and Wyldstyle, as well as featuring the Twelfth Doctor and the TARDIS.
Finally, the Meet That Hero! instalment Supergirl Meets E.T. featured Supergirl (DC Multiverse) using Wyldstyle as an example of someone who couldn't tell that Superman and Clark Kent were the same person due to him wearing glasses as a disguise. At the end of the video, the Twelfth Doctor appears and presents his TARDIS, with E.T. (E.T. the Extra-Terrestrial) noting its external appearance ("Phooooooone.").
The theatrical film The LEGO Batman Movie features licensed use of the Daleks among the various villains from the Phantom Zone; the film was later adapted into game form in a LEGO Dimensions story pack. Additionally, The LEGO Movie 2: The Second Part features a brief appearance by a LEGO TARDIS.
Other connections
To be added