Nail polish actually seems to have been normalized by the early 1880s: trade magazines and books of “practical hints” were sharing information about manicures and varnish. Generally these were whitish/clear (intended to cover smoothed and trimmed nails), but they could be tinted a bit pink.
From New York’s Great Industries, Exchange and Commercial Review (1885):
Mrs. Mary E. Cobb, Manicure, Nos. 64 and 66 West 23rd Street. [Big premises!] – Among the many professions which have won their way into popular favor is that of the manicure. … Mrs. Mary E. Cobb,whose elegant parlors … have been the resort of some of our best and wealthiest families. The business was established in 1878, and occupied No. 69 for a number of years, it having been removed to its present more commodious quarters quite recently. Mrs. Cobb is a lady of rare artistic ability and talent, whose skill as a manicure has helped to beautify many of our most fashionable belles, and the preparations sold by her are the very best of the kind made. Among the principal ones are bran-u nail powder, the most brilliant finger nail polish in the world. Cosmetic “cherri-lip” for tinting finger nail, lips and cheeks produces a rosy blush true to nature.
By 1893, druggists’ formula-books were carrying multiple recipes to make unbranded nail polish powders. Generally, these were tin oxide with scented oils, sometimes with carmine to make it a bit pink. Liquid nail enamel seemed to come about around this time; “Diamond Nail Enamel” was a common name, and it was frequently advertised along with polishing powders, tinted lip/cheek salves, ink-stain removers, etc. Druggists’ recipes for nail enamel/varnish around the turn of the century included paraffin, chloroform, and scented oils – not tinting materials at this time, though they appeared in “fashionable shell-pink” by the mid-1920s, by which point nail polish powders were done and “nail polish” meant what it does today.
By 1932, nail polish was being made in multiple shades of red/pink, which could be chosen by the wearer to harmonize with her clothing or lipstick. During the 1930s, it was most common for a manicure to be done with opaque white polish over the tip of the nail and the half-moon on top of the red/pink base, but by the early 1940s this was passé, and polish was applied as we do today.
As far as "wildly popular" goes, that's a pretty subjective way to put it - I'm not sure I would describe it as ever having become wildly popular. It was a successful product from the late Victorian period, though, and as they made the innovations described above, the newer version became more popular than the old.