Eton had the birch in use right up until the 1970’s at least, so it sure wasn’t there.
I suspect this is going to be roughly related to Gladstone’s Universal Education Act, which was around 1874. Once education to the age of 13 became mandatory for both sexes, something easier to wield and keep in storage was needed than a birch, which, for a start, needs about 8 or more individual rods or shoots to make up a whole birch.
Plus, as females were now to be compulsorily educated, something that could be applied to bare hands rather than bare bums was needed. What Victorian pater familias did or paid people to do to their daughters at home was another matter, but NOT in the schoolroom, thanks!