The Programme

Yes, the programme is titled: "Doctor Who", but from the very beginning
– in fact from before the beginning, the title has also been written as:  “DR WHO”.

And as to the central character ...

Just as with the programme itself -  from before the beginning, the central character has also been referred to as: "DR. WHO".


[From the same original proposal document:]

In the BBC's own 'Radio Times' TV guide (issue dated 14 November 1963), both the programme and the central character are called: "Dr. Who"

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On page 7 of the BBC 'Radio Times' TV guide (issue dated 21 November 1963)
there is a short feature on the new programme:

Again, the programme is titled: "DR. WHO"

This time, however, in the text of the article itself, the character is also referred to as: "the Doctor".

So, it would seem that, from the outset, both: "the Doctor" and: "Dr.Who" were correct ways of referring to the character.

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In the same issue of the BBC 'Radio Times' TV guide - in the actual listings for 23 November, 1963 [the first ever Episode], again we see the programme titled as "DR. WHO", and William Hartnell's character is also listed as: "Dr. Who"

And so it would be continuously throughout the years up into the fourth incarnation of the central character.
Although he was primarily referred to as "the Doctor" in-episode, he was referred to as: "Dr. Who" in the episode listings:

Also, for the first eighteen years of the programme's history (1963 - 1981), in the closing credits for the first four incarnations of the Time Lord, the character is listed as either: "Dr. Who" or: "Doctor Who"  - not as: "The Doctor"

In fact, it is not until the beginning of the fifth (Davison) incarnation that the character is listed in the credits as: "The Doctor" - something which would also continue through the Colin Baker and Sylvester McCoy years, up to the suspension of the programme.

When the programme returned to television in 2005, the character was once again listed as: "Doctor Who"

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In the final episode of the Eccleston era: "The Parting of the Ways", the credits include:

"and introducing DAVID TENNANT as Doctor Who"

It was only in the Tennant era proper, beginning with the Christmas Special: "The Christmas Invasion", that the character was again listed as:
"The Doctor"

It seems fairly clear that it was acceptable, for those actually working on the programme, to refer to the character as: " Dr. Who".

In an interview promoting the story: "The Three Doctors" (December '72 - January '73), Barry Letts, the programme's then producer, explains "The new story is that there is a black hole in space through which cosmic energy is draining. Everybody is helpless. Even Dr. Who - if he is alone. So the only people he can draw on to deal with it are his other selves."
And Verity Lambert,  the programme's original producer, states: "I cast William Hartnell as Dr.Who, after seeing him playing a rugby talent scout in the Richard Harris film This Sporting Life."  

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In fact, right up to the end of season 18, the character was still officially being referred to as: "Doctor Who":

[Production paperwork for the story: "The Keeper of Traken" ]

Admittedly, season 16 had seen one story: "The Power of Kroll" where the production paperwork identified the character as "The Doctor".
Then in season 18, three of the seven stories also saw the character listed as "The Doctor" ["The Leisure Hive", "Full Circle" & "Logopolis"]
Yet it was only really with the last of these - the very last story of season 18 and the final story of Tom Baker's tenure - that the character reference formally changed to: "The Doctor" in the production paperwork.

And even then, there would be a small handful of stories where - in the production paperwork - the character would still be identified as: 'Doctor Who'
[i.e. "Four to Doomsday", "The King's Demons", "Resurrection of the Daleks", "Time and the Rani" & "Battlefield"]

Also, from 1966 to 1986, Annuals were produced to accompany the programme. [No annual was produced for 1972]

All of the annuals from 1966 to 1979 were titled: "THE DR WHO ANNUAL". Only in 1980 did the title change to: "DOCTOR WHO".

The second annual (1967) simply states:
"As played by WILLIAM HARTNELL", and the annuals for '68, '69, and '70 state: "As played by PATRICK THROUGHTON".

However, all of the annuals from 1971 to 1981 clearly refer to the character as: "DR WHO"
The Annuals for '71, '73, '74, and '75 bear the wording: “Starring JON PERTWEE as DR WHO”.
Those for '76, '77, '78, '79, '80, and '81 - "Starring TOM BAKER as DR WHO”

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.And of course, there were also the Doctor Who novelisations, beginning in 1964 with:

[Paperback Edition]

."Dr WHO in an exciting adventure with the Daleks"

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This was followed in 1965 by: "Doctor Who and the Zarbi", and: "Doctor Who and the Crusaders"

These were reprinted in 1973, and in the years from '73-'94 they were followed by somewhere in the region of 150 further novelisations - of which, 66 out of the first 70, had titles clearly referencing the Time Lord as:
'Doctor Who'
 
Doctor Who and the Auton Invasion
Doctor Who and the Cave Monsters
Doctor Who and the Doomsday Weapon
Doctor Who and the Day of the Daleks
Doctor Who and the Dæmons
Doctor Who and the Sea-Devils
Doctor Who and the Abominable Snowmen
Doctor Who and the Curse of Peladon
Doctor Who and the Cybermen
Doctor Who and the Giant Robot
Doctor Who and the Terror of the Autons
Doctor Who and the Green Death
Doctor Who and the Planet of the Spiders
Doctor Who and the Loch Ness Monster
Doctor Who and the Dinosaur Invasion
Doctor Who and the Tenth Planet
Doctor Who and the Ice Warriors
Doctor Who and the Revenge of the Cybermen
Doctor Who and the Genesis of the Daleks
Doctor Who and the Web of Fear
Doctor Who and the Space War
Doctor Who and the Planet of the Daleks
Doctor Who and the Pyramids of Mars
Doctor Who and the Carnival of Monsters
Doctor Who and the Seeds of Doom
Doctor Who and the Dalek Invasion of Earth
Doctor Who and the Claws of Axos
Doctor Who and the Ark in Space
Doctor Who and the Brain of Morbius
Doctor Who and the Planet of Evil
Doctor Who and the Mutants
Doctor Who and the Deadly Assassin
Doctor Who and the Talons of Weng-Chiang
Doctor Who and the Masque of Mandragora
Doctor Who and the Face of Evil
Doctor Who and the Horror of Fang Rock
Doctor Who and the Tomb of the Cybermen
Doctor Who and the Time Warrior
Doctor Who and the Android Invasion
Doctor Who and the Sontaran Experiment
Doctor Who and the Hand of Fear
Doctor Who and the Invisible Enemy
Doctor Who and the Robots of Death
Doctor Who and the Image of the Fendahl
Doctor Who and the War Games
Doctor Who and the Destiny of the Daleks
Doctor Who and the Ribos Operation
Doctor Who and the Underworld
Doctor Who and the Invasion of Time
Doctor Who and the Stones of Blood
Doctor Who and the Androids of Tara
Doctor Who and the Power of Kroll 
Doctor Who and the Armageddon Factor
Doctor Who and the Keys of Marinus
Doctor Who and the Nightmare of Eden
Doctor Who and the Horns of Nimon
Doctor Who and the Monster of Peladon
Doctor Who and the Creature from the Pit
Doctor Who and the Enemy of the World
Doctor Who and An Unearthly Child
Doctor Who and the State of Decay
Doctor Who and Warriors' Gate
Doctor Who and the Keeper of Traken
Doctor Who and the Leisure Hive
Doctor Who and the Visitation
Doctor Who and the Sunmakers

And then of course, in London's newly-completed G.P.O. Tower: