theories on the origin of the Welsh hat

There is no clear indication of how the Welsh hat appeared as it did, but here are some theories:

Some women in France at the beginning of the 19th century wore tall hats, a little like the Welsh hat. We know that Thomas Price (Carnhuanawc), a close friend of Lady Llanover had contacts with Bretons during the late 1820s and early 1830s; some Welsh hats have ‘Paris’ labels and there is a poem which blames the French for tall hats and this has been put forward as evidence that the Welsh hat had a French origin, but so far, there is no firm evidence of this. (Ken Etheridge, Welsh costume, (1958), pp. 43-46) Drawings of this type of hat in France are not common but it is possible that the ‘Paris’ labels were used by British manufacturers to indicate the type of manufacture or to inflate prices.

A watercolour of a Tyrolean woman wearing a hat very similar to a Welsh hat was copied by Princess Victoria from an original by her drawing master in 1832 while she was staying in north Wales. (Marina Warner, Queen Victoria’s sketchbook, (1979), p. 18), and George Burrow described a hat worn by a woman living near the Devil’s Bridge in 1854 as wearing ‘a high conical hat like that of the Tyrol’ (George Borrow, Wild Wales, 1860). There are other examples of illustrations Tyrolean hats which are very similar to Welsh hats – for example a selection of 1450 costume drawings originally published in Munchener Bilderbogen, 1861 – 1890 and selected by Braun and Schneider, Historic Costume in Pictures, by Dover Publications Inc., (1975)

By the end of the 19th century, the Welsh hat was being dismissed as part of the traditional Welsh costume, mainly on the grounds that it didn’t appear before 1800. Its origin was sought elsewhere:  some thought it was a survival from Puritan times:

Even the so-called Welsh hat, which was still to be worn in the sixties by women in Cardiganshire, less frequently in Merioneth, Carnarvonshire, and Anglesey, has nothing distinctly Welsh. It was introduced from England, as may be seen from the examination of paintings, dating from the Stuart times. John Rhys and David Brynmor-Jones, Welsh People, (1900), p. 567

Some thought it was introduced by the Flemmings:
Those hats were imported from the Lowlands at the beginning of the fifteenth century, but enterprising tradesmen of recent times had tried to make people believe that they were an emblem of Welsh nationality, which was ridiculous. Report on Professor Timothy Lewis’ lecture to the Carmarthen Cymrodorion on “Cymru pan oedd Cymro’u Frenin” (Wales when the Welshman was King), Cambrian News and Merionethshire Standard, 28 January 1916

Others thought it came from Spain. A French tourist used the word bolivar for hats worn in Holyhead market in 1880, from the Spanish meaning top hat with wide brim, like a Spanish sombrero.

1854 Bangor
The costume of the Welsh peasantry and common people is awkward and strange. The old women wear large round men’s hats. Our guide said, that in the old wars—which were so desperate and bloody, in defence of their wilderness mountains and rocky fastnesses—the women fought as did the men, and that they continue to wear their hats in memory of their prowess and powers of self-protection.
Benedict, Erastus Cornelius, A run through Europe, (New York, 1860), p. 543

During the Royal visit to Swansea in 1881:
The front two ranks of the choir were filled with between 90 and 100 young girls dressed in the ancient Welsh costume of plaid skirt, bodice, and shawl and the ‘cockle hat’ so called from its resemblance to the shell fish … some of the damsels wore for their head gear the tall, conical and broad-brimmed hat generally supposed to be the typical head covering of the Welsh but which antiquarians assert is a Spanish innovation. The Bristol Mercury and Daily Post, October 19, 1881

According to a suggestion made by Mr Silas Evans, before his death, that 500 of the choir should be dressed in Welsh costume, a large number of the choir were so attired, the head gear varying between the old, tapering Welsh tall hat, the Spanish style, broader at the top, and the “cockle” hat. Cardiff Times, 22.10.1881

The only hat which is of the same shape may be found in Korea, but it is very small and is worn by men:
A Merthyr man writes to point out the striking similarity of the Corean hat to the sugar-loaf headgear of Welshwomen, and he wants to know whether this does not prove our Eastern origin. Evening Express, 28 November 1894