The Neville possibility was first mooted in 2005 by Brenda James, who later collaborated with William Rubenstein on a book, The Truth Will Out. Others, such as Dr John Casson, have furthered the case.
The main plank for Neville as Shakespeare is that his educational, cultural and political background, and time line of travels and experiences almost exactly match the Shakespeare plays' chronology and themes. As well as extensive real-life experience in this Shakespearean "world", he also had demonstrable writing and scholastic skills, a personal library, and disposable time and income, in other words, things that the "real" Shakespeare seemed to lack.
While Neville's political roles still allowed time for other pursuits, he was also aware the topics and themes of his plays could jeopardise his career if true authorship was known (his involvement with Essex nearly cost him his head, as it was). Even after his death, his family may well have shared the same sensitivity.
While there is no direct evidence how the London Shakespeare came to "front" his plays, Neville was a known mutual associate of playwrights such as Jonson, Fletcher and Beaumont.
No "smoking gun" as yet, but it's still early days in this new line of research. There has long been speculation that a few pages of script of a play about Sir Thomas More is the only surviving "Shakespeare" handwriting. Some contend this matches Neville's handwriting. Notations and references in his journals, letters and personal library books are also apposite to various "Shakespeare" works. Two Shakespeare major verse works are dedicated to Neville's friend and Tower companion, the Earl of Southampton. Space precludes mentioning many other links.
But if Neville colluded with London theatre man Shakespeare to "front" his writings, where does that leave Stratford Will? I believe previous attempts to reconcile the few pieces of evidence connecting the Stratford man to London Will have been misguided. Here I refer to the Blackfriars property documents, the London deposition signature (both identifying the signatory as from Stratford-upon-Avon), and the small bequests in his will to the three London theatre men. Other scenarios simply assume these few items prove Stratford William and London William to be one and the same.
I have another explanation as to why they weren't; that of sheer coincidence, the basic essential ingredient for major mysterie since time immemorial. Neville may have been the Main Man, but that still leaves two other Will Shakespeares.
Next week's denouement speculates how they all got so mixed up. And it's all to do with a boyhood neighbour of the Stratford Will who went to the bright lights of London as a printer's apprentice.