Matters of originality aside, Doctor Who played a key role in traumatically effecting my obsession with the weird. At the age of about five, I went through that classic rite of passage common to many children of a certain age living amongst the dreary desolation that was 1970s Britain: of being scared witless and hiding behing the sofa (or settees as we called them then) whilst watching that classic horror tv series, Doctor Who. Yes, for me at the time Doctor Who was a horror series. Unsurprising given that my earliest memory of the programme was during Jon Pertwee's tenure, when the Doctor and Jo Grant were rowing through a sludge of fucking green luminescent slime (which would kill you horribly if you touched it) in a mine cart whilst being threatened by giant fucking maggots! Not only did this traumatise my five year old self, but laid a firm foundation upon which was built the gateway into a strange new world of the horrible and the fantastic. Happy days!
But I get ahead of myself. Going back to the origins of Who, there is not much to be said by way of Lovecraftian themes or influences in the First and Second Doctor eras (William Hartnell and Patrick Troughton). As a horribly mutated and tentacular species inimical to all life, the Daleks probably come closest to being Lovecraftian: 'Exterminate!!!' being possibly the defining epithet of a Lovecrafitian (or perhaps more properly Ligottian) universe. The Hartnell era also firmly established a conceptual link to Lovecraft with regard to what he considered to be a driving of the aesthetics of the weird - namely the 'adventurous expectancy' that arises from the imagined suspension of the natural law allowing one to participate in infinite voyagings across time and space. In this respect, I think we disregard Lovecraft'a emphasis on awe and wonder at our peril: the perception of him being first and foremost a writer of horror (rather than of the weird) has become far too overdetermined. In a twist on Peter Cannon's The Lovecraft Chronicles, one could almost imagine in an alternative timeline an older Lovecraft playing the First Doctor (quite possibly wearing his father's old suit):
The First Doctor?
As other's have noted, Brian Lumley's later Titus Crow stories seem to be a thinly-veiled Derlethian spin on Doctor Who, with Crow jetting about time and space in de Marigny's clock (hence the title of this series and, if I recall aright, also bigger on the inside). Indeed, I have a recollection of the Master's TARDIS during the Fifth Doctor's era taking the form of a grandfather clock...
With the second Doctor, what we have by way of Lovecraftian
stylings has largely been retconned by later Who (particularly in the New Adventures novels of the 1990s - more on those later), such that the Great
Intelligence has been identified in some Whovian sources Yog Sothoth. It remains unclear as to whether this is Whovian 'canon' or not, although intriguingly the most recent series featuring Eleventh Doctor Matt Smith has the
Great Intelligence as, quite literally, an Eater of Souls. This is, of course, one of Yog-Sothoth's sobriquet's, introduced in the Illuminatus Trilogy in the cosmic entity consumes and entraps the life-force of human beings like
flies in amber for all eternity. This is also the version of Yog Sothoth that
that Charles Stross works with in his chilling short story A Colder War. However, there is much speculation here - although there is one instance during the reboot of Who when 'the Great Old Ones' are actually mentioned...
As noted above, the entry of Pertwee's Third Doctor marked (at least for me) a tonal shift towards the horrific, which would later reach its apex
in the classic Robert Holmes stories of the Fourth Doctor's era. Though ostensibly
no explicit Lovecraftian references here, we do have the tentacular Axons from
The Claws of Axos, and of course the Silurians and the Sea Devils, who come
closest to an overt nod of the head to Lovecraftian themes: ancient, prehuman
races - akin to the serpent men of Valusia and the Deep Ones - returning to
wipe out humanity.
Even so, both of these stories were tempered by shifting societal attitudes during the late/post-hippy era as the Doctor sought
to broker a peace between these ancient races and humanity - although in both
cases ineffectively in the face of the blunt militarism of the British
government and UNIT. As with the Great Intelligence, the Nestene intelligence
of the classic Auton stories has also been retconned as the Great Old One Shub
Niggurath in the New Adventure novels. In addition, at least one
rpg iteration of Doctor Who has also tried to link the Sea Devils to
Lovecraft's Dagon. Another story echoing notably Lovecraftian themes is, of course, The Daemons, where ancient supernatural evil turns out to be the intrusion of extraterrestrial forces. The Three Doctors also introduced the protoplasmic Gell Guards...
as well as Omega: a figure from Gallifrey's ancient history - a history which, in the Doctor's later incarnations, would be revealed as replete with intriguing Lovecraftian possibilities (the forbidden knowledge of the Black Scrolls of Rassilon...).