As in title above, at what playing strength level should one be able to mate with King, Bishop and Knight vs King without much effort (with reasonable time left for the game)?
A member of my club, rated about 1900 FIDE, had this in a championship blitz game about 2 years ago with 30 seconds left on the clock. He delivered mate with time to spare. IMs have failed. Go figure. If you study and learn then you should be able to deliver mate. If you don't study then you won't. Playing strength doesn't have that much to do with it.
I don't think it's that relevant; most chess players are unlikely to ever get this endgame on the board. I'd rather spend my time studying other tricky pawnless endgames like Queen vs. Rook, which is far more likely to occur.
I knew how to win this endgame fairly early in my career, but I had always been fascinated with endgames; conversely, Wikipedia shows two grandmasters failing to deliver mate. Had they known the right corner for mate, and the W-manoeuvre, they'd probably figure out the rest themselves.
In 2007, Jeremy Silman published Silman's Complete Endgame Course: From Beginner to Master. In this book, endgames aren't organised by theme but according to the level at which they would be most useful to learn - at least according to Silman. The book is divided into chapters that progress from the beginner level (below USCF class E) to "master" (rating range 2200 - 2399). In the introduction to the book, Silman states (page xv; emphasis added):
I've deliberately left out many endgames. Why? Because I don't feel they are important to players below the 2400 level. For example, I heretically decided not to include Bishop and Knight vs Lone King because it is far from easy to master, and it occurs very rarely in over-the-board play. In fact, I only got it once in my entire career, while IM John Watson and IM John Donaldson never got it at all!"
So he concludes that this endgame is so rare that it isn't worth spending two to three hours of study on it if your level is below 2400.
Not learning this endgame before the 2200 level is an idea that has been criticised by many people below the 2200 level who have faced this endgame on the board. (A long time ago, I was in this situation myself in a rapid game in a school tournament. As luck would have it, I had studied that endgame just the night before, so when my opponent demanded a draw, not knowing it was a losing endgame, I protested and checkmated his king, which he kindly moved to the edge of the board...) For example, this thread on the Chess.com forums has both detractors and defenders of the approach.
Yuri Averbakh's Erfolg im Endspiel (translated into German by M. Hermann, Sportverlag 1987) is a book aimed at the elementary to intermediate level (which are not clearly defined in the book's introduction) that treats this endgame in two different chapters: in the second chapter (on elementary checkmates of the lone king), starting from positions where the lone king is just one square away from the last rank or file, and again in the last chapter, starting from positions where the lone king is either in the wrong corner or farther away from the edge (e.g. sixth rank). This same chapter also covers endgames with two knights versus pawn, rook and bishop versus bishop, and rook and knight versus rook. This approach recognises that this is a rather difficult endgame when starting with the lone king near the middle of the board, while at the same time treating it as an endgame that needs to be learnt well below the 2200 rating level.
Max Euwe six-part book series Praktische schaaklessen (published in Dutch in the early 1980s) covers the endgame both in the second volume (which covers each phase of the game) and the sixth volume, Hogeschool van het eindspel ("Endgame Academy"), which treats endgames in greater depth.
The endgame is not explained in The Soviet Chess Primer by Ilya Maizelis (translated into English by John Sugden, 2014). The book's chapter on endgames is obviously not meant to be exhaustive; the endgame type that is treated in most details is rook-and-pawn endgames.
At the bottom line, there is no consensus when the Bishop-and-Knigh-versus-Lone-King endgame should be learnt. In Studying Chess Made Easy, grandmaster Andrew Soltis said that he had never encountered this endgame in his own games but said that "learning it teaches techniques that can be applied elsewhere" (quoted from Wikipedia).
One should learn chess from the ending first! So if one takes your question as when one should have learned the technique then as a beginner. If one has never looked at this endgame (shame on you) and are facing it for the first time in a real game with the 50 move rule and clock ticking then anything can happen.
The minimum one should know is quite small and is just a couple key concepts. Delétang's Triangle Method is easy to remember. It's just 3 successive nets.
I coached a high school team with ratings from 700 to 1500, and demonstrated the B + N Mate and how you move the opponent's king to a corner whose color is the same as the squares on which your bishop travels.
Those around 700-800 had trouble with it, maybe because they were fairly new to competitive chess, but those 900 or better could all do it.
I think you should be able to teach a six year old who doesn't know how all the pieces in chess move to do it, since there's a structure to how you do it.