The standard answer to this is the late nineteenth century
, when Delia Bacon’s book, The Philosophy of the Plays of Shakespeare Unfolded, claimed that the Shakespeare canon was not written by William of Stratford, but was the result of a collaboration of a team of courtly writers led by Francis Bacon. This, however, was only the moment when the issue was opened to the reading public at large, for the issue itself has been there ever since the first peeps of Shakespeare criticism. As Albert Feuillerat explains in his Composition of Shakespeare’s Plays (1953), the question of the authenticity of the Shakespeare canon
has been raised with more or less insistence ever since the eighteenth century. . . . Pope conjectured that in Love’s Labor’s Lost, the Winter’s Tale, The Comedy of Errors, and Titus Andronicus there was nothing authentic except a few scenes and some characters (1725). . . . Similar doubts were expressed by Theobald regarding Henry V (1734), by Hanmer regarding the Two Gentlemen of Verona (1744), by Samuel Johnson regarding Richard II (1765), and by Farmer regarding The Taming of the Shrew (1767). Ritson found some disparities so evident that in The Two Gentlemen, Love’s Labor’s Lost and Richard II he claimed he could distinguish Shakespeare’s hand as easily as one could recognize the brilliant brush strokes with which a Titian might have sought to touch up a mere daub. Malone in 1790, in his often quoted dissertation on Henry VI, did not recognize Shakespeare’s hand except in some passages of the second and third parts and thought that the first part came entirely from one of Shakespeare’s predecessors. (32)
The difference between these early questioners and Delia Bacon is that they never disputed the existence of a William Shakespeare as author, however sparing his touch. As lawyers and doctors began openly questioning the Stratford biography, Frederick Fleay and the so-called disintegrators got ever more severe in limiting the evanescent Shakespeare’s involvement in the production of the canon. Clinging like drowning survivors of shipwreck to that crumbling bit of flotsam, the name itself, academics and their groupies continue to defend what, if one takes the long view, never really existed. The first public attribution, by Francis Meres in 1598, is hardly solid since it stands alone while the book that introduces Shakespeare’s name to the reading public was his only connection with the world of poetic literature. The other attribution, that found in the First Folio of 1623, is fragile in the extreme, and nothing since has done anything to solidify it, quite the reverse. We’re left with Authority’s age-old pronunciamento: “It’s so because I say it’s so.”
That someone wrote the magic and that during the 1590s the name William Shakespeare got attached to it along with a good deal else that’s questionable is all we can be certain of, and all that anyone could be certain of for a very long time. Beginning with Delia, the search began to replace the name with that of a writer whose biography made more sense, keeping Shakespeare only to identify the canon, as in A.W. Pollard’s article of 1917: “Shakespeare’s fight with the Pirates,” in which by Shakespeare he meant, not the author, but the Lord Chamberlain’s Men and their precious playbook.
The questioning has taken new turns over the years. At the beginning it must have been about who was writing the plays that began to be performed in the early 1590s. This was answered in 1598 when three of the most popular plays were published as by William Shake-spear (Richard III) or Shakespeare (Richard II and Romeo and Juliet), the same time that it (the name) was introduced to the reading public via the Meres book as the author of several other plays as well, no doubt popular plays, and of certain “sugar’d sonnets.” Some readers were already familiar with the name from the title pages of two narrative poems published four and five years earlier, Venus and Adonis and The Rape of Lucrece.
Thus towards the end of the 1590s, long after the first versions of some of the plays had been seen by audiences (the Contentions and True Tragedies), they had a name for the author, but they never had the man himself. The lack of any solid history of William’s presence in London, of letters to him from other writers or from him to other writers or from anyone to anyone else about him, suggests that the questioning must have continued, which the ambiguous wording of the front material in the First Folio was intended to put to rest. That this wording continued through later editions suggests that the questioning continued until, as Feuillerat reports, the emphasis began to shift to questions about what seemed to be other hands of far less ability.
Long story short: the authorship question has been around from the beginning, it has simply shifted focus repeatedly from one aspect to another. Where it rests at the moment, on which of several candidates actually wrote the works attributed to the Stratford money-lender, is only one stage in the long ongoing question of who actually wrote the canon, and, perhaps most important, why it’s taking so long to come up with an answer.
politicworm 
To me, it is hard to argue that Shakspere did not spend time in London. The tax authorities were looking for him there. His neighbor sent him a letter there. The 1612 Mountjoy information. The 1616 mention by Jonson of him acting in plays around the turn of the century. His name is associated with the Globe and Blackfriars gatehouse properties. Some of his small claims seem to have been filed in London. Where would he and his father have gone to get their arms approved.
Which of these is the fishiest?
I didn’t say he didn’t “spend time” in London, what I said was that there’s no solid history of his presence there, which there isn’t. Yes, a room was rented in his name in two places, perhaps three, where obviously he never spent enough time to make an impression on anyone who mattered. The only moment when he can actually be found in London was to sign a statement for someone whose room he rented once who was involved in a lawsuit; when he was needed for a second deposition, he failed to show up. Many people who lived within a two day journey from London had business there. His father probably had business there and probably he was sent there on business, and probably, once his name began appearing on plays, he went to find out if perhaps he couldn’t parlay that into something even better. But the immensely gifted playwright for the leading theater company that performed at Court would have lived for many years as close as possible to the theaters where his plays were being produced. Were this true of William we would have evidence of it, as we have of Ben Jonson.
Still, your question deserves a more complete answer. Perhaps in a future blog.
As I argue in my book, and have argued at Wikipedia, there is credible evidence (subject to some interpretation, see Gibson’s The Shakespeare Claimants) that contemporaries Hall and Marston considered ‘Shakespeare’ to be a cover-name for a hidden author, possibly Bacon. This clearly goes beyond doubt about certain portions of the canon and puts considerable stress on the Stratford theory.
I believe the first publicized mention of ‘doubt’ was in Hart’s The Romance of Yachting, 1848, with predates Delia Bacon, though there is recently a new development on this. I’ll have to look it up.