Everyone catches the revision of "fifteenth" to "eighteenth" from Josephus' Jewish War (1.401) to his Antiquities (15.380). No one seems to catch the significance of the other revisions. In War, Josephus literally said Herod restored (epeskeuasen) the Temple that year. Obviously no one thought that was true, but in correcting himself, Josephus also takes pains to convey a more nuanced process. Now he says Herod "undertook (epebaleto) an extraordinary work, (namely) the reconstructing (kataskeuasasthai) of the temple of God".
Note that Wikgren's Loeb translation made an infinitive into a gerund. Interestingly, Whiston did not: "undertook a very great work, that is, to build". If we isolate this sentence, Whiston seems awkward and Wikgren's decision is justifiable, but in the larger context, Whiston underscores a key point, and Wikgren, although inadvertently, has misled us.
The other critical detail here is that "undertook" does not exactly mean "began". The verb (imperfect passive form of epiballw, taking the accusitive case) more literally means Herod had it thrown upon him, which suggests something like Liddel & Scott's alternate glosses for "undertook", which are, "took (or put) it upon himself". Essentially, the desire has taken firm root, but there is no implication that the intended action has necessarily been embarked upon, as of yet.
In the War, Josephus told us one year in which Herod built. In the Antiquities, Josephus corrects this with exacting qualification. He now tells us only what year Herod devoted himself to the building project. Every detail of the narrative following bears this out. Herod's offer to begin the project was met with skepticism by Jews who feared he might tear down and not build up again. So the King promised "he would not pull down the temple before having ready all the materials" (15.390) and Josephus concludes that Herod indeed, "began the construction [note the same root in kataskeuhs] only after all these preparations had diligently been made by him."
The materials included "a thousand wagons to carry the stones" and the preparations included the training of "a thousand priests" as masons and builders. There is no telling how long it took to train a thousand priests into skillful laborers. There is no telling how many trips the thousand stone wagons took, before enough stone was piled up at the site to begin tearing down... which tearing itself may not even have been considered as the beginning of "reconstruction".
The only thing we can date to 20/19 BC, according to Joesphus, is the speech in which Herod promised to build. The actual building must have begun quite some time later. One or two or even three years is not an unthinkable amount of time for the immense amount of preparations that had to take place before reconstruction could begin. The tearing down would probably have been very quick, so the rebuilding could have begun in 19, 18, or perhaps early 17 BC.
Josephus later says the Temple sanctuary was completed in "a year and six months" (15.421) but this by itself does not contradict anything else. We still do not know how much time passed after the speech before work on that new sanctuary was actually begun. However, we also know that shortly after this eighteen month period Herod visited Caesar in Rome (16.6). Since Caesar went north into Gaul in 16 BC, Herod can only have sailed to Italy in 18, 17 or early 16. The latest possible date for sanctuary construction to begin would therefore be winter of 18/17 BC. The earliest, we should think, not before 19. Most feasibly, it could have been anywhere in the 24 month window of 19 to 18 BC.
The point of all this - for New Testament chronologists - is that these references from Josephus are not enough, by themselves, to inform us precisely about what year the Jerusalemites were speaking in when they told Jesus, "This Temple was under construction (oikodomhthe) for forty-six years". Without inventing a time span for the prep-work, that "46 years" could have ended in 27, 28, 29 or even early 30 AD (counting strictly or inclusively).
Therefore, John 2:20 does provide us with a tight range of dates, but not one so restrictive that it should have daunted apologetic concerns in the past. Naturally, these considerations renders moot a vast chunk of everything that has been written by apologists about John 2:20 and Josephus' Antiquities 15, before now.
*Post title partly in homage and congratulations to Jona Lendering for his forthcoming publication (in Dutch) about Common Errors in scholarship.