For everyone else the role of wizard is held a priest.
The greenskin have their shamans as their wizards
The ogres have their butchers who are wizard-cook-priests and the firebellies who are just priests
Dwarfs don’t have priests and therefore no wizards
Chaos Dwarfs DO have priests and those guys are wizards
In Nehekera lich priests are wizards and as clearly evidenced by their name priests
For Lizardmen their wizards are the Slann (Priestly caste), and skink priests.
Skaven are half and half, in that technically they have other wizards but their main WIZARD guys (Grey seers) are also priests.
Is it because elves were the only ones with a secular magical tradition and the empire got their magic tradition from elves.
Cathay also have secular mages, and outside of the regular 8 winds they also use high and dark magic, often at the same time (though i think they need to be shugengan, that is dragon descendant, for the high and dark), so their magic lore and knowledge are more advanced than the Empire and more diverse than the high elves.
just goes to show you what happens with dedicated state investment
They also turned the Chamon lore into alchemy advanced enough to be on the brink of real chemistry.