Pretty much what it says on the tin, when did the shift for most people calling Britain The United Kingdom actually happen? I've been reading primary sources from between world war I & II and noticed that Most authors were just referring to it as Britain or Great Britain, But now it seems like most people say the United Kingdom.
Britain and the United Kingdom are different political entities. The UK's full name is the "United Kingdom of Great Britain and Northern Ireland". Great Britain is England, Scotland and Wales.
The two terms are often used interchangeably but the difference matters if it is important to explicitly include/exclude Northern Ireland. Unionists in NI typically regard themselves as British (not Irish); Republicans in NI typically regard themselves as Irish (not British).
Whether the shift in usage you have noticed is real, I don't know. But the conflict in Northern Ireland, which reignited in the late 1960s and came to an uneasy end with the Good Friday Agreement in 1998, did educate a lot of people about the difference. One of the provisions of the GFA was that citizens of NI could choose to be British, Irish, or both, including in their choice of passport(s) to hold.
You don't state your nationality or the nationalities of the authors you are reading. There may be country-specific differences in how usage has changed.