Yes, this was 100% the real life reason for the accents. Roddenberry was trying to show a very diverse crew that could have international appeal. He did it in a way we’d find ham-fisted today, but for his time it was very progressive.
Picard does speak with a French accent in Star Trek:Picard while pretending to be a villain. It’s a good thing that was just one scene.
And, of course, James Doohan wasn’t actually Scottish (he was a Canadian, born to Irish immigrants), and while Walter Koenig was of Russian Jewish descent, he was born in Chicago.
In this season of Strange New Worlds, Scotty is finally actually being played by a Scottish actor.
Simon Pegg played Scottie in the “Kelvin Timeline” films by JJ Abrams, but he’s an English actor.
Also in the Kelvin films, Chekov was played by Anton Yelchin, who was born in the Soviet Union. However, he immigrated to the US with his parents when he was six months old, so he grew up completely in the US. He probably based his accent on the way his parents spoke. So maybe a bit more authentic.
What he did is beautiful, and it still gets me misty eyed even in objectively bad episodes of Trek. I didn’t watch TOS but this progressive ethos carries in some way to all other Trek series. He invented a positive vision of the future that included everyone. And that still has resonance today.
It’s pretty unique among futuristic sci fi stories. Almost everything else is dystopian in some way. Star Trek shows a future where so many problems are solved. It’s not perfect, of course, or it would be a boring show, but it’s still a rare vision of hope.
Easy explanation: The Second Hundred Years War was won by the English. 
He cosplays as a Scot, though…
Actually, he has said “half my family is Scottish” in the past. Although he may just be talking about his wife.
To explain Picard’s accent, I just assume that France got around finally to surrendering to the British in the Hundred Year’s War.
[tea snob alert]
I had some of the best tea of my life in France, specifically Mariage Freres. They have a lot of high quality tea, including their French Breakfast, but the Milk Oolong is truly sublime.
[/tea snob alert]
Seconded. I still have some of their Christmas blend
Also, the pastries … L’Étoile mystérieuse is one I still dream about.
Palais des Thés has also had boutiques for decades, there are tea salons… it is not like tea is unobtainable or superlatively expensive in France (also, you can order it off the Internet, like everywhere) but the question is, would a random cafe have anything passable or just a few bags of floor sweepings?
I do not have the latest statistics, but the number of French who do not drink tea should be in the minority these days. That was not always the case, e.g. in the 1990s. So who is to say tea will be regarded as in any way exotic in the Star Trek future? (especially with cheap transporters making import trivial, at least on Earth—Terran tea might not be readily available on Romulus)
Plus, matter replicators, such as the ones on the Enterprise from which Picard was ordering his “Tea, Earl Grey, hot,” which even eliminate the need for transport of things like tea leaves.
Kinda like this? 
from xkcd: Period Speech.