Here is an answer based on the TV series. The 7th Doctor serial Remembrance of the Daleks is set in 1963, the year in which the first episode of Doctor Who was broadcast. It contains a scene in which an announcer on a television says:
This is BBC television, the time is quarter past five and Saturday viewing continues with an adventure in the new science fiction series Doc—
but someone switches the television off before the sentence is finished.
This isn't definite proof of course - it could be that Doctor Who takes place in an alternate universe where there was a different new BBC science fiction series in 1963 whose name began with "Doc" - but it does suggest that the show Doctor Who exists in the Whoniverse, in the 1960s at least.
On the other hand, though, one of the early episodes of the Christopher Eccleston era has a character search the internet for "doctor blue box." He comes up with lots of conspiracy stories about a mysterious time traveler, but nothing about the TV show Doctor Who. This suggests that in the New Who continuity the TV show doesn't exist. I guess we'll have to put it down to disruption of the space-time continuum due to the time war or some such. (Long-term continuity has never been a strong feature of Doctor Who.)