This answer is going to focus predominantly on Europe, with an emphasis on womenswear.

There is little evidence of true pockets being in use before the 15th Century, and even the ones that where used from the 16th to the 19th centuries in womenswear are probably not what you would envision today! But people have always needed to carry their things around with them, and up until the end of the medieval period, both men and women would usually carry a pouch of leather, skins or cloth to carry tools, coins or whatever they needed. It would be often fastened to whatever belt they wore; Ötzi the Iceman lived 10,000 years ago but was found with a leather drawstring purse sewn to a belt, containing flint tools, a bone, and some fungus that may have had fire lighting properties.

Throughout the Medieval period in many parts of Europe, people of both genders carried leather pouches on their belts. Both men and women would wear a tunic, with men’s to the knee and women’s to the ankle, belted with a leather belt upon which tools could be hung. The simple pouch on the tunic is the origin of pockets.

From the 13th century, slits called fichets started to be cut in the super tunic to allow access to coin purses hanging from the belt on the tunic underneath. These weren’t lined like modern pockets, just slits you could reach through. In the 16th century, pockets grew in popularity and were hung from a belt around the waist. Women would conceal them under skirts, but men would just wear them under coats, outside the rest of their clothing.

Lucy Locket lost her pocket... how do you lose a pocket? Well, in the 17th century, men could enjoy sewn in pockets but women could not until the late 19th century. Women would wear a shift, stays (which especially in common women were more like a bra than a corset in that they provided support rather than restriction), at least one petticoat but usually more, a bodice and an over skirt. Skirts were very voluminous and easily hid bulky pockets. Styles of bodices, over skirts and gowns varied, but underwear was functionally the same until drawers became popular in the latter half of the 19th century. Yes, women really went commando! But back onto pockets. They became, by the 17th century, a pair of deep decorated pouches sewn onto a ribbon or cord, which was tied around the waist over the petticoats but under the skirt. They would be decorated with brocade, ribbons and embroidery if the woman could afford it. This example is typical of an 18th century pair of pockets. The gown or over skirt would have a slit cut in the sides like the supertunics of the 13th century with their fichets, so women could reach though for their things.

The 19th century was a period of transition, with both sewn-in and tie-on pockets in use for women. Sewing patterns from the 1840s show pockets sewn into the side seams similarly to how they are today, although they’ve got smaller as fashions got tighter.

Ultimately, pockets grew out of the pouches which would be more comparable to a small modern handbag. The modern handbag developed as skirts became too slim for large pockets to be concealed underneath from the 1790s onwards, at the same time that sewn-in pockets began to exist. Pockets in womenswear have been influenced by fashion as well as practicality since at least the 15th century, which may contribute to why they are so small in modern women’s jeans!

TLDR: “Lucy Locket dropped her bumbag” would be equally accurate, albeit less poetic.