I would have thought that more than one copy of this pamphlet has survived and it is highly probable that at least one copy exists somewhere in a UK Government collection, given what was seen as the significantly seditious nature of the work and the recent memory of "The War of American Independence", the French revolution and of course the Napoleonic Wars which were still in full flood.
I suspect you are right. Since the pamphlet identifies the author only as "A Gentleman of the University of Oxford," archivists and librarians would have had no reason to treat it as something special. I suspect that a few libraries are combing through their inventories right now. Let's hope.
Apparently the TLS took the position that they weren't going to fuss over a poem when no one knew if it was any good, a rather narrow view, I would say. It's Shelley's, and his sometime collaboration with Elizabeth (I think he wrote with her before) is surely of interest even if it's unlikely to be an undiscovered classic. Of course, we'll never know, etc.
Narrow, yes. But reasonable, too. There's so much undistinguished polemical verse from this period, even from fine poets. It might be stirring; it might give insight's into the development of Shelley's radicalism and as a poet; or, it might be something whose sole value comes from the fact that "Shelley" wrote it. I vote for "Wait and See."
Meanwhile, there are two undeniable aspects to this story:
-- on the positive side for all of us: the literary public is getting involved, which will put pressure on the owner and/or future owners to treat this responsibly
-- also: the dealer is getting an awful lot of publicity, always good news for those whose goal is maximizing the sale price.