Other mistake: Crowley shows acute discomfort while inside the church with Aziraphale and the Nazi spies as he is on consecrated ground. However, (spoiler alert) once the bomb destroys the church he is perfectly comfortable. The key word here is "ground" - it is the land upon which the church stands that is consecrated, not the building itself. (This allows churches to be rebuilt if they are demolished or destroyed.) Crowley should still be in considerable discomfort after the church has been blown up.
Factual error: In order to take delivery of the demon baby, Crowley meets Hastur and Ligur - two fellow demons - in a church graveyard, but later we find that demons cannot step on to consecrated ground without suffering extreme discomfort. Church graveyards are always on consecrated ground.
Factual error: Aziraphale cites the only prophecy from Agnes Nutter that he could find as one for 1972, "do not buy Betamax" (a reference to the home video format war). Betamax was only launched in 1975 starting in Japan, so this advice should have been for then, not 1972. These prophecies were completely, even ludicrously accurate, so that excludes any explanation that she was simply a few years off, because she was never wrong. (00:19:10)
Other mistake: When Aziraphale is reading The Book in his shop, you see various written prophecies pass by on screen. These are the original 1990 versions, not those actually read out later in the show. E.g. "When Orient's chariot inverted be, four wheels in the air" vs "When Robyn's chariot inverted be, three wheels in the air" and the inclusion of The Other Four Horsemen of the Apocalypse, who were cut from the show.