Note: The OP's linked tune on Wikimedia is not the Simon & Garfunkel tune. It's nowhere near, totally different melody.
For any traditional song you need to source several versions & derive a history to decide where copyright, if any, rests.
Wikipedia claims that the Simon & Garfunkel version came from Carthy, via a book by Ewan MacColl, The Singing Island (1960), which transcribed a performance by a Durham miner, Mark Anderson (1874-1953) in 1947.
As no recording exists of the Anderson version, this book would be considered the first hard reference of the version that became the Simon & Garfunkel song; and would be your source reference to compare any changes.
The Martin Carthy version bears much closer resemblance to the Simon & Garfunkel than the traditional tune in the question, so this would form a good middle-ground for comparison to the MacColl/Anderson version. This, though, would also give Carthy copyright on the intermediate changes - this gets very complicated.
That would really make Anderson the owner of any copyright of this set of derivative versions only - in the same way Alan Price has copyright to the Animals version of House of the Rising Sun, credited Trad:Arr Price - though as Anderson apparently never did anything further with it, even if contested by his heirs that copyright will expire next year [author's death + 70 years].
I would consider the song itself to be all but public domain, so long as you avoid any of the elements added by Carthy/Simon/Garfunkel.
I have found a hugely comprehensive work-through of the various versions over time [will take quite some time to study thoroughly].
"...Tell Her To Make Me A Cambric Shirt"
From The "Elfin Knight" to "Scarborough Fair"
From this is would seem that the Wikimedia version is referenced in this as 'Whittingham Fair' From the Newcastle Courant, 1879, which is a century out of copyright.
This reference contains transcriptions of both the Carthy and MacColl variants.
To directly answer your question, you are quite at liberty to write completely new lyrics [or even derivative if your source pre-dates Carthy/S&G] so long as the tune you use is also copyright-free.