The BBC's First World War drama 'Birdsong' had Collared Doves played in the background, during scenes which took place in England, before the war. Somehow all the more amusing to find a birdsong related historical inaccuracy, considering the title of the thing!
Practically every scene in The Crown that's set at Balmoral in the 70s and 80s features a cutaway to a Red Kite. Unless there was some kind of top secret re-introduction programme that predates the 1989 one, this is also an example of historically inaccurate birds. I'd audibly scoff every time I saw this, much to the annoyance of my partner.
Nightingales in America is a particular bugbear in TV. Also, the first episode of The Last Kingdom shows a falconer handling a Harris Hawk in early-medieval Northumbria. Of course, the misplaced calls of Great Northern Divers is the most prolific example in TV and cinema. Hearing it over the crater-scarred battlefield of the Somme in the film 1917 is a stand-out for me.
Obviously none of this really matters, but it can take you out of it for a moment (and annoy me, in particular).
I have to say though, it does increase my respect for production teams when they include geographically correct bird calls. For example, in Queen's Gambit. Lots of Yellow Warblers in the scenes set in the United States and Coal Tits for the scenes in Russia. Lots of kudos from me for that kind of attention to detail!
Even wildlife documentaries have been known to get this wrong. The Attenborough series Seven Worlds, One Planet was called-out for featuring the call of the Red-legged Seriema for a bit in a forest 1,000 miles from where the species actually occurs. Tut, tut!
For an example from "reality" TV, how about the Great Crested Grebes and Eurasian Black Vulture shown in the last series of the BBC's Race Across the World while the contestants travelled across the mountains of Chile?
As you can tell, I've dedicated way too much attention to all this (and also am possibly watching too much TV...).