The notion that none always takes a singular verb is a grammatical myth. For a rejoinder, we need only quote Strunk and White: "None are as fallible as those who are sure they're right." The summary that Merriam-Webster's Dictionary of English Usage gives on this point nicely encapsulates the consensus position of most usage guides, including even very conservative ones like Fowler's:
Clearly none has been both singular and plural since Old English and still is. The notion that it is singular only is a myth of unknown origin that appears to have arisen in the late 19th century. If in context it seems like a singular to you, use a singular verb; if it seems like a plural, use a plural verb. Both are acceptable beyond serious criticism. (p. 664).
What I find interesting about the exchange in this video is not so much the fact that Fry is wrong but the rhetorical strategies that he uses to browbeat Davies into accepting his position, because they are typical of people who try to foist their incorrections off on others.
First, Fry invokes the word's meaning: "None means 'not one'....Not one of them works. You can't say, 'One of them work and one of them don't.'"
There are two ways to understand this argument. First, we could understand this as an appeal to the word's etymology, which would be a classic example of an etymological fallacy. True, the word comes from Old English ne + an (not one), but not only does the origin of a word not control its present usage, but even in Old English, the word could be plural.
If he's not arguing from etymology, then he's arguing from a paraphrase of the word's meaning. That too doesn't tell us anything about the grammatical status of the word itself. To see why the argument fails, consider a parallel argument for police, which by the same logic, we must conclude should be singular:
"Police means 'a group of law-enforcement officers.' You can't say, 'A group of law-enforcement officers work.'"
That's utter crap, of course. Police is plural no matter what the status of its paraphrase. Further, if we paraphrase none differently, for example as "no people" or "no things," we get a different number from "not one."
The next strategy Fry tries is analogy: " It's like people saying 'between you and I,' thinking they're being impressive. You say 'between you and me.'"
This is a flawed analogy on two levels. First, case agreement (I vs. me) is not the same as number agreement, so whatever the status of case marking in Standard English, it's irrelevant to understanding the number of none. Second, Fry cites between you and I as an example of a hypercorrection, but does he seriously think that plural none is a hypercorrection? It's an ordinary part of colloquial English. People don't use plural none to try to sound more proper. They use it unreflectively, as part of their ordinary usage.
The rhetoric here appears to be one of authority. Fry cites an unrelated usage point to buttress his credibility, in the hopes that if Davies accepts the one, he will accept the other.
Davies does not simply roll over on Fry. In his own way he points out that none can have plural signification with an example: "All those batteries on the table, they're all dead. None of them work."
At this point Fry inserts, "...is incorrect." In other words, he is reduced to dogmatic assertion.
Unfortunately, given the confident demeanor of Fry, most viewers will be left with the impression that he's correct. Certainly, that was the opinion of the poster of the You-Tube video, along with many of the commenters. It is pitiful that a shoddy, illogical argument completely contrary to current or past usage and unsupported by reputable style guides should pass for grammatical truth, but this, alas, is the state of most grammatical commentary.