The Netflix series "Rise of Empires: Ottomans" suggests that he was definitely Scottish (even giving him a Scottish accent and style) - and this show is attempting to show an accurate portrayal of events.
However, on researching this, it seems the only evidence of him being Scottish is an unsubstantiated claim by the Historian Steven Runciman. It seems more likely he was German.
Has there been any further evidence to suggest he was actually Scottish?
He's described in the contemporary sources as "German" and a member of the Genoese defenders led by Giovanni Giustiniani.
The Byzantines sometimes called everyone from the west "Germans", so he wasn't necessarily German and could be from some other country. But in this case, the source isn't a Byzantine Greek source. He's described as a German by Leonard of Chios, who was ethnically Greek, but was a Latin Catholic priest, educated in Italy, and he certainly would have known about the different nations in the west.
In Latin, Leonard calls him “Johannis Grande Alemani”. Maybe he’s simply describing him? Maybe he was tall, or big - just a big German guy? It would be strange for a German person to have the name “Grand” but he could have been from France or Italy, especially from eastern parts of France or northern parts of Italy that were part of the Holy Roman Emperor and could have had Germans living there.
Steven Runciman is the only source for the idea that he might have been Scottish…but Runciman never said why he thought so. Grant is a popular Scottish surname (although ultimately French in origin), and he’s described as an engineer, and Scottish people were famous for being engineers in the 18th and 19th centuries, and Runciman’s ancestors were Scottish. So maybe he just made it up! All that depends on reading "Johannis Grande" as "John Grant" in the first place.
But Runciman was extremely influential, so it’s just sort of become a well-known fact, in academia and in pop culture, despite the complete lack of evidence.
Sources:
Steven Runciman, The Fall of Constantinople, 1453 (Cambridge University Press, 1965, repr. Canto, 1990)
Marios Philippides and Walter K. Hanak, The Siege and the Fall of Constantinople in 1453: Historiography, Topography, and Military Studies (Ashgate, 2011) (even they still refer to him as a Scot sometimes!)