This phrase, “the fullness of time,” meaning, more or less, the appropriate time, was originally confined to Christian contexts. For example, in the King James Bible, Galatians 4:4 reads, “But when the fulness of the time was come, God sent forth his Son, made of a woman, made under the law.”
In due course — or in the fullness of time — the expression began to universally take the form “in the fullness of time,” meaning at the appropriate time, or after a certain amount of time, usually lengthy, has passed. The first secular use cited by the OED unsurprisingly is from Charles Dickens (Barnaby Rudge, 1841), who, as he did, brought the high-flown rhetoric down to earth: “Nor was she quite certain that she saw and heard with her own proper senses, even when the coach, in the fullness of time, stopped at the Black Lion.”
As far as transatlantic patterns go, Google Ngram Viewer shows much more frequent use in the U.S. than in Great Britain in the late nineteenth-century; I’d venture that the reason is America’s greater degree of public piety.
The phrase’s popularity shot up in Britain between 1900 and about 1945, corresponding, I’d say, to its adoption there in non-religious contexts, especially favored by windbag politicians, and those making fun of them.
But in the fullness of time (sorry, can’t help myself), America caught up. Reliable figures for Ngram Viewer only go up to 2000, but the Corpus of Global Web-Based English — a snapshot or nearly 2 billion words on online text in 2012-13 — shows nearly equal use of the phrase in the U.S. and U.K.
Since it began publishing in 1865, the New York Times has used the phrase 210 times, but 17 percent of them have been since 2010. For example, in reference to astronomical shifting, science writer Dennis Overbye observed in 2018, “In the fullness of time, everything gets everywhere.” And that same year, in a review of George Bernard Shaw’s St. Joan, theater critic Jesse Green wrote:
“What other judgment can I judge by but by my own?” Joan asks, casually assuming supremacy over churchmen and kings. From this idea comes not only the necessary sentence of a perfectly fair proceeding but also, in the fullness of time, Protestantism, nationalism, individualism and, as Shaw would have it, the Great War, which had recently concluded as he started writing the play.
notoneoffbritishisms.com 
I think I have only encountered the phrase ‘In the fullness of time’ in the context of people being intentionally pompous, in a mocking way.
This is from the TV series ‘Yes, Minister’.
I thought of Yes, Minister straight away as well.
That is a series that people all over the world should watch, as it is the greatest satire on the practice of politics that you will ever see.
I was disappointed not to find it in one of the wryly admonishing sections of Fowler, such as ‘Hackneyed Phrases.’ However, these phrases come and go, and when Fowler was writing (or the Fowlers were writing), it might have gone. Such phrases are often revived, then reinvigorated, in offices. ‘Not in my gift’ and ‘Not in my bailliewick’ were very popular some years ago. I think that this phrase had the same treatment, and might even have been seen in the company of ‘At this moment in time’.
Dennis Overbye’s usage (“In the fullness of time, everything gets everywhere”) seems to mean “eventually” or “by the end of time” more than “at the appropriate time”.
For me, like a couple of others, this term is associated with Yes, Minister. An aunt (ANT) introduced me to the show in the mid-90s and I was completely taken with it, since it replicated almost perfectly my own experience of– at that time– a decade in the (US) federal government. I appropriated the expression and used it frequently over the next 20 years of my career– in all seriousness or to joke about the frustrations of government’s, well, deliberate pace.
I have heard, or read, that phrase occasionally in the past, but it was brought back to me in a HUGE way when Queen Elizabeth said it very recently, like maybe 2 weeks ago, about when Charles will become King. She said it will be in the fullness of time.