Professional magician, member of Magic Circle.
I’m not aware of any definitive answer to this question. However, there are specialists who, while perhaps not performing magic very much, study the history of magic in great detail, and one of these historians may be able to provide a better answer.
We know that there have been magicians and mystics of various types more or less as far back as records go. We know that they have employed numerous different aids to misdirection, of which sticks and staves and wands are just one type, albeit one that has achieved some prominence over time.
As regards these sticks and wands, we know that there have been many different types and designs, and that they can be traced back to many sources. Alchemists used sticks and rods to stir their chemical concoctions. Many types of staves and rods have been used as weapons, and sometimes these have become modified over time into ceremonial sticks, rods, batons and so forth, some denoting rank, status or power. Any of these and more could, over time, have been adopted by different types of wonder-workers and have evolved into the generic wands we often see associated with magic today.
French maestro magician Robert Houdin is often regarded as the father of modern magic, and it may be that he used a ‘classic’ wand modelled on the walking canes used in France in his day.
But that’s all I’ve got. Perhaps the truth will never really be known.
I’d also like to point out that even today, there are many different types of wands. Not that many magicians actually use the ‘classic’ (black, white tips) wand you’re asking about, and countless magicians never use any sort of wand at all. My field is mentalism (mind-reading and psychic-flavoured magic) and mentalists never use wands of any description.
However, while not all that many magicians use the ‘classic’ wand in real life, this is the design employed by cartoonists, illustrators and others who need quick visual shorthand for ‘magician’. In other words, it is more common as a convenient symbol than as an actual real-life prop. In similar vein, fewer than 1% of magicians ever produce a rabbit from a top hat, but it’s still useful as a handy piece of visual shorthand.