The excellent coverage of this question at The Phrase Finder points out an earlier (than 1732) source that conveys the gist of the saying in a translated Spanish proverb:
Captain John Stevens's work, 'A New Spanish and English Dictionary', published in London in 1706, translates a Spanish proverb, as "Do not leave off your Coat till May be past".
But then the author of this article observes:
Those rhymes may well have originated in England and migrated across the Channel. It is difficult to understand why the Spanish would coin such a proverb, which would seem a little cautious for that part of the world - the average temperature in Seville in May is 20°C.
The Phrase Finder's author ultimately votes firmly in favor of the end-of-the-fifth-month-of-the-year interpretation of the saying over the hawthorns-past-their-bloom interpretation:
All in all, although the May blossom interpretation seems appealing, the 'May' in this proverb is the month of May.
Jennifer Speake, Oxford Dictionary of Proverbs, Fifth Edition (2008), cites both Stevens and Fuller in her brief coverage of the saying—and then offers this interesting quotation from Robert Graves, The White Goddess (1948):
In ancient Greece, as in Britain, this [May] was the month in which people went about in old clothes—a custom referred to in the proverb 'Ne'er cast a clout err May be out,' meaning 'do not put on new clothes until the unlucky month is over.'
So the Graves theory seems to be that the May referred to is indeed the month of May, but that the point of not casting off a clout during May has to do with an old English (and ancient Greek) superstition about not prematurely donning new clothing during the month of May.
Followup: Thanks to the excerpt that jlovegren provides from a different edition of The Oxford Dictionary of Proverbs (and to Mari-Lou A's asking about it), we can see that an earlier version of the Spanish form of the proverb has existed at least since 1627—but instead of saying "Until May has passed..." it says "Until May..." That suffices to get the citizens of Seville out of their overcoats on May 1, and it gives Spain a 105-year head start in documented existence of the proverb over the first native English occurrence. It begins to look to me as though Spain has the stronger claim.