The earliest reliable reference I’ve been able that definitely refers to the use of a cane in an English school comes from Rugby c.1828. The birch was also still in use at Rugby at that date (and for a considerable period after that as well). It wasn’t that uncommon for both methods to be in use at a school – as was the case at Eton throughout much of the twentieth century.
But it was definitely in use well before that – a poem called ‘The School-Master’s Cane’ was published in England in 1787 and I’ve found references to it (as distinct from the birch) as early as 1777.