You see this all the time in fiction, Man in the High Castle, Wolfenstein, Hearts of Iron, but basically the Nazis take over America (or other countries) and that countries flag suddenly just has a swastika in it. For America it’s usually the 48/50 stars being reorganized into a Swastika shape but everything else looks the exact same.
Does this have any basis in reality, the Nazis changing their occupied territories flags to include Nazi symbols?
I don’t know about the Nazis, but other countries did:
Here are flags of Axis powers and their allies:
Not many of the nation states succeeding the British Empire would previously have been identical nation states with the same boundaries and state structures, and from what I can tell from digging around the internet, whatever flags and symbols used simply disappeared from view once the states/kingdoms in question (e.g., Moghul Empire, Indian princely states, Burmese and African kingdoms) were absorbed into the Empire. Those flags incorporating the Union Flag were wholly new inventions, as are their successors.
The hammer and sickle is probably a closer analogy than the Union Jack - many USSR states had it in their flags/national emblems (or allusions to it, like the GDR). Now, you could argue those were all part of the same union, and it’s one of the international symbols of communism, but nevertheless it wouldn’t have been on a lot of those flags without Russian hegemony.
Right, the fictional universes mentioned in the OP operate under a notion that the specific cult-ideology of Nazism and the form and structure of the NSDAP as ruling party would be adopted whole hog in the conquered nations, often all the way down to using German titles. That the “nazified” America government would be run by the American Nazi Party and the Bund as we knew them rather than be taken over by, say, the KKK with their affectations.
IRL the various fascist/collaborationist movements in non-German lands tended to adopt their own symbols and their own ideological denominations. And as MrDibble points out, in the communist countries not absorbed by the USSR you more often got things like the red star, or localized analogues of the H&S for representing labor or revolution (not to mention places like Czechoslovakia, Cuba and Poland where the flag stayed free of party symbols).
Along a similar line it has and is still going on here in the United Sates with state flags showing smybols of the Confederate States of America.
Wikipedia:
It suits both puppets and puppeteers for puppet states to have an outward veneer of legitimacy. The de jure rulers of the puppets are of the nationality of the puppet country, so wish to appear to have their independence, even if they have a similar ideology to their de facto masters. It suits the puppeteers for their puppets to outwardly appear to have their own free choice, rather than it being plainly obvious that the puppets are having the puppeteer’s will imposed from outside.
For colonies, and things like the post-confederate states, their rulers would share nationality or identity with the parent state and wish to preserve that through the use of familiar parent state elements in their flags and so on.
Another example is the 11 countries with a crescent moon on their flag.
This indicates a religious (Muslim) empire in addition to a political one.
For that matter, a large number of European flags were originally crusaders’ crosses. The Scandinavian flags still are. The British flag is a combination of the English and Scottish crusaders’ crosses. Many provincial flags still have medieval crosses on them.
However all that said, once the Nazi’s have conquered the known world I can easily imagine them demanding encouraging vassal states to put Nazi party stamp as a demonstration of unity and allegiance.
IIRC, Hitler and the Nazi leadership didn’t think much of the local Nazis in their occupied territories, and mostly kept them on a short leash or even out of government altogether. Eventually they seem mostly to have used them to recruit for the Russian Front.