The traces of Besson or Ramelli within early English literature are faint.
John Evelyn (1620-1706) briefly mentions Ramelli in his "Numismata, a discourse of medals, ancient and modern" (1697). Is his list of "Renowned Architects and Sculptors" he notes the greats: "M.Angelo, Primaticius, Pantormo, Iohn de Bollogne, Francis Flemingo", Bernini, "Vignole, Scammozzi, Alberti, Del'Orme, Serli, not forgetting Iohn de Vdine, Inventor, or Restorer of the Art of Stucco." To this list he adds "Other skillful Achitects and Mechanicks, Bellou, Ramelli, Caus, Zonca, &c. whose Books are commonly adorned with their Effigies" (pg. 147)
While Evelyn certainly had a passing knowledge of Besson and Ramelli, John Wilkins (1614-1672) seems to have had a deeper familiarity with their work. In "Mathematicall Magick" (1648) he gives both Ramelli and Besson high praise by favourably comparing them to Aristotle:
"And whereas the Mathematicians of those former ages, did possesse all their learning, as covetous men doe their wealth, only in thought and notion; the judicious Aristotle, like a wise Steward, did lay it out to particular use and improvement, rightly preferring the reality and substance of publike benefit, before the shadows of some retired speculation, or vulgar opinion.
Since him there have been divers other Authors, who have been eminent for their writings of this nature. Such were Hero Alexandrinus, Hero Mechanicus, Pappus Alexandrinus, Proclus Mathematicus, Vitruvius, Guidus Vbaldus, Henricus Monantholius, Galileus, Guevara, Mersennus, Bettinus, &c. Besides many others, that have treated largely of severall engines, as Augustine Ramelli, Vittorio Zoncha, Iacobus Bessonius, Vegetius, Lipsius.
Most of which Authours I have perused, and shall willingly acknowledge my self a debtor to them for many things in this following Discourse." (pg. 7)
Wilkins mentions Ramelli again in a footnote, referring to the device on "Fig. 160" (pg. 91). The inclusion of this figure is fitting given the title of the chapter "Of the Wheel, by multiplcation of which it is easie to move any imaginable weight." Four pages later, he also cites Besson with the rather uninformative: "Stevin. de Static. prax. See Besson." (pg. 95)
Earlier I had speculated about the possibility of Besson's influence on Hooke in his creation -- with Boyle -- of the air pump. I had dismissed the possibility based on the date on which he acquired his own Spanish copy of the work. It is possible, however, that he may have had access to it through Wilkins. As a youth, Hooke attended Westminister School under the tutelage of headmaster Dr. Richard Busby. As described by Cooper (2004), Busby -- a friend of Wilkins -- allowed Hooke to skip class to spend time among London's artisans to develop his mechanical talents. His talents attracted Wilkins's attention:
"When Wilkins heard of Hooke’s precocious talent he presented him with a copy of Mathematicall Magick. Hooke could then see that the abstractions of Euclidean geometry, at which he excelled, were useful when designing and making machines, for which he had a passion. Mathematicall Magick and its author had a great influence on Hooke’s intellectual development. A copy of the book, probably the one presented to him by Wilkins, was in Hooke’s library at the time of his death." (pg. 50-51)
Hooke certainly would have been acquainted with the names of Besson and Ramelli through Wilkins and it's possibly that he may have had access to the actual works before starting work with Boyle.
Cooper recounts another interesting detail from the diary of Samuel Pepys. On May 1 1665 a carriage containing Royal Society members Hooke, Wilkins, Lord Brouckner, and Sir Robert Moray pulled up along side Pepys. As a newly appointed member, he joined them for a trip to the country to see Colonel Thomas Blount's new chariot. They left Blount's house and carried on to Deptford via Greenwhich to visit Sayes Court, home of John Evelyn. We can only imagine if Hooke, Evelyn, and Wilkins spent the afternoon wandering the grounds and discussing the work of Besson or Ramelli.
Cooper, Michael (2004 May/June). Robert Hooke (1635-1703): Professional scientist, engineer and surveyor. Ingenia. 19: 48-54.