Wands were invented in Europe, but we don't know who first used one.
The only real mention of the origin of wands is in the Pottermore writing on Uagadou, which is the African wizarding school.
The wand is a European invention, and while African witches and wizards have adopted it as a useful tool in the last century, many spells are cast simply by pointing the finger or through hand gestures.
Nowhere is it mentioned who actually invented wands, or who had been the first to use one. However, if we believe the Pottermore writing, both the inventor and first user of a wand (who could easily be the same person) would have been European.
It's certainly not Ollivander or Gregorovitch - wands are too old.
We can say for certain that wands existed before Gregorovitch or Ollivander could have invented them. This is shown not simply on Pottermore, but in the books themselves as well as the early supplementary books. The sign on Ollivander's shop says they've been making wands since 382 BC.
“The last shop was narrow and shabby. Peeling gold letters over the door read Ollivanders: Makers of Fine Wands since 382 BC. A single wand lay on a faded purple cushion in the dusty window.”
Harry Potter and the Philosopher's Stone, Chapter 5 (Diagon Alley)
A crowd chases a Quidditch team that beat their home team while wielding their wands in 1385.
“Quidditch was well established in Ireland by the fourteenth century, as proved by Zacharias Mumps’s account of a match in 1385: ‘A team of Warlocks from Cork flew over for a game in Lancashire and did offend the locals by beating their heroes soundly. The Irishmen knew tricks with the Quaffle that had not been seen in Lancashire before and had to flee the village for fear of their lives when the crowd drew out their wands and gave chase.”
Quidditch Through the Ages
Salazar Slytherin had a wand, which is stated on Pottermore to later be owned by Gormlaith Gaunt and Isolt Sayre, so wands would have existed in the Founders' time. It had been passed through the centuries to Isolt, who lived in the early 1600s, so wands existed centuries before that.
In all the years that she had lived with it, Isolt had never known that she held in her hand the wand of Salazar Slytherin, one of the founders of Hogwarts, and that it contained a fragment of a magical snake’s horn: in this case, a Basilisk. The wand had been taught by its creator to ‘sleep’ when so instructed, and this secret had been handed down through the centuries to each member of Slytherin’s family who possessed it.
Ilvermorny - (Pottermore)