This page has a depiction of a wizard from a late 19th century lantern slide:
It sort of seems like this guy is somewhere between the modern Merlin/Gandalf vision of a wizard and older depictions of the "renaissance magus" like the one shown here of Dr. Faustus from a 1620 edition of Faust:
And here is an illustration of Merlin by Gustave Doré from 1868, you can see the image of the wizard already included the robes and the long white beard but perhaps the conical hat wasn't yet part of standard wizard regalia in the public imagination:
And this illustration of Merlin from the 1889 edition of Mark Twain's A Connecticut Yankee in King Arthur's Court has a short conical hat, but again not the really tall kind we now associate with wizards:
I would guess that "The Sorceror's Apprentice" from Disney's Fantasia helped cement the modern iconic image of a wizard's hat (in the same way Wizard of Oz cemented the image of a witch, inventing the notion that they had green skin for example). In this cartoon, Mickey gained temporary magic powers from "borrowing" the hat of his wizard master, which was pointed, blue, and had the star-and-crescent moon pattern often seen in later images of wizards. Fantasia came out in 1940, only 3 years after The Hobbit, so I don't know how likely it is that the Disney animators were influenced by Gandalf (especially since according to the "Design and Animation" section of the Fantasia wiki article, "Animation on The Sorcerer's Apprentice began on January 21, 1938", and this article says "Houghton Mifflin Co. of Boston and New York published the first American edition of The Hobbit in spring of 1938 following its September, 1937 debut in the United Kingdom from George Allen & Unwin LTD").
This page has some of the story of how Disney artists came up with that particular design for the hat:
For the actual filming of the animated classic, a well-known silent film star, Nigel de Brulier, was brought in to provide live-action reference footage for the character of the sorcerer. Disney writer Carl Fallberg went to the famous Hollywood costume rental house, Western Costume, to find the appropriate robe and pointed hat for the performer to wear during the shooting. Fallberg had to paste white stars and crescent moons onto the hat to get the look he wanted.