As I found by accident, Nedham may have been thinking of the very passage from Machiavelli that Farneworth used this phrase in.  “Che il tempo a consumare i desideri della libertà non bastiè certissimo: perché s’intende spesso quella essere in una città da coloro riassunta che mai la gustarono, ma solo per la memoria che ne avevano lasciata i padri loro la amavano, e perciò, quella ricuperata, con ogni ostinazione e pericolo conservano; e quando mai i padri non la avessero ricordata, i palagi publici, i luoghi de’ magistrati, le insegne de’ liberi ordini la ricordano: le quali cose conviene che sieno con massimo desiderio dai cittadini cognosciute. Quali opere volete voi che sieno le vostre che contrappesino alla dolcezza del vivere libero, o che facciano mancare gli uomini del desiderio delle presenti condizioni?”

Here’s an anonymous translation of 1901.  I’ve changed the boldface phrases to something more literal.  “That time can neither destroy nor abate the desire for freedom is most certain; for it has been often observed, that those have reassumed their liberty who in their own persons had never tasted it, and love it only from remembrance of what they have heard their fathers relate; and, therefore, when recovered, have preserved it with indomitable resolution and at every hazard.  And even when their fathers could not remember it, the public buildings, the halls of the magistracy, and the insignia of free institutions, remind them of it; and these things cannot fail to be known and greatly desired by every class of citizens.  What is it you imagine you can do, that would be an equivalent for the sweetness of living free, or make men lose the desire of their present conditions?”

(The translation has “tasted of its charms” where I put “tasted it,” and “the sweets of liberty” where I put “the sweetness of living free”.)

Ellis Farneworth’s translation of 1775 used Nedham’s phrase, “the sweets of freedom,” but not where Machiavelli mentioned sweetness.  “That time is not able to eradicate the love of Liberty, is sufficiently evident; since it has often happened in States where the citizens themselves were not free, that many have exerted their most strenuous endeavours to be so, merely upon the report of the blessings of Liberty, which they have received from their fathers, and when they succeeded, and tasted the sweets of freedom, have despised all difficulties and dangers to maintain it....”

Farneworth says the first English translation of this work was in 1675, that is, after Nedham.