I have also found a reference to the cane in the 1820s, at St Paul’s School, London. Perhaps surprisingly, it is made clear that this was administered to the hands. This is in a letter to The Times from an Old Boy of the school some decades later. I’ve temporarily lost it but when it turns up I’ll put it on my website.
Assuming St Pauls was not completely untypical of boys’ schools, my working assumption is that the cane was regarded then as a fairly minor punishment, the birch being reserved for more serious matters. So it would be “as well as”, not “instead of”. Later in the 19th century, as the birch became less acceptable in some quarters, the cane evidently replaced the birch for more serious punishments, but in most such schools administered to the seat of the trousers instead of the hands (and probably with more strokes as well.)