Bet you didn't know this - there's still an extremely small portion of people in UK who use "thou", "thee" and "thy" in casual speech. It's exceptionally rare, rural, and is considered a moribund thing to do since the only people who still do it are in their last years, and probably don't do it all the time anymore. Ben Weatherstaff from The Secret Garden would have been a dying breed even in 1911 when the story was published (turns out I was totally wrong about this! Keep reading). In his broad Yorkshire dialect, the words come out "tha" (thou and thy sound the same) and "thee".

Update to this topic: until recently I discovered a movie called Kes, a more modern-day example of 2nd person singular pronouns being used in a movie that takes place - and is shot completely in - broad, rural Yorkshire dialect as it existed in the the 1960's. "Tha" and "thee" pronouns abound, and the amazing thing is that it is all spoken in a completely natural manner. So it sounds completely natural! Most actors using "thou" in works like The Secret Garden have, I think, a tendency to deliver the archaic lines in such a way that puts a sort of undue focus on their archaism. They think that because the word is archaic, it naturally should be delivered with an archaic quality, but that's not natural speech. Kes's cast consists of people who are all from Yorkshire and natively speak the broad dialect, so it sounds like they're actually speaking it, not acting it. By contrast, The Secret Garden's actors make the old-fashioned lines sound more dated than they apparently were.

So, it seems, I was wrong about my assumption that 1911 already saw the general ousting of "thou" in daily use. Quite the contrary, if you follow Kes as an example, "thou" was used clear into the 60's!