First of all you would need to listen to track 30 on the Topic album MY FATHER’S THE KING OF THE GYPSIES: MUSIC OF ENGLISH & WELSH TRAVELLERS & GYPSIES TSCD661 to hear the original of Levi Smith's Georgie to realise what a brilliant job Martin's interpretation of that performance was and how he made an accompaniment that allowed Levi's free-time singing to flow without being straight-jacketed by a rhythmic acompaniment. However, I would need to be convinced what connection there could be between a 20th century English gypsy and a medieval troubadour in their approach to music.
The author (Ian Pittaway?) assures that he has made huge assumptions and I reproduce the words that he puts in bold type:-

So if troubadour songs were sung freely, and if this was the reason they continued to be written non-mensurally when rhythmic notation had become common, then the impossible exercise of trying to standardise notation for Martin Carthy’s performance may be a parallel example to illustrate the liberty with which troubadour songs were performed, and may possibly be the reason they continued to be written in non-mensural notation.

These and other considerations reduce the impact of what is nevertheless an interesting hypothesis.
I am reminded of an experiment that a group of top jazzmen were asked to make in the 1950s. In the first decade of the 20th century, the leading figure in the early days of jazz in New Orleans was the trumpeter Charles Joseph "Buddy" Bolden. Photos of him exist but no recordings. The group had access to lots of information in terms of accounts of other jazzmen who had heard Buddy and were influenced by him as well as lots of early jazz recording, They had all the information, all the musical skills but their efforts foundered on just how much the music managed to 'swing', how the early syncopations would have manifested themselves, how technically adept his band musicians would have been. Even with all their skills they did not feel that they they could reproduce the rhythmic norms of half a century before. If that was the case, what chance has anyone with much less information available to reproduce the style and approach of musicians a half a millenium ago?