Machiavelli's treatise The Prince, written in the early 16th century and first published in 1532, was placed on the Index of Prohibited Books in 1559. According to Robert Bireley, quoted on Wikipedia,
there were in circulation approximately fifteen editions of the Prince and nineteen of the Discourses and French translations of each before they were placed on the Index of Paul IV in 1559, a measure which nearly stopped publication in Catholic areas except in France. Three principal writers took the field against Machiavelli between the publication of his works and their condemnation in 1559 and again by the Tridentine Index in 1564. These were the English cardinal Reginald Pole and the Portuguese bishop Jerónimo Osório, both of whom lived for many years in Italy, and the Italian humanist and later bishop, Ambrogio Caterino Politi.
By 1559, when the Index was created, the Church of England had already been separated from Rome for 25 years, so one would not necessarily expect a ban imposed by the Pope to be relevant in England. However, according to a footnote in Drama in English From the Middle Ages to the Early Twentieth Century (edited by Christopher J. Wheatley; Catholic University of America Press, 2016),
[The Prince] was officially banned in England in 1559, and the first printed translation in English was not published until 1640. Especially those who had not read him, [Machiavelli] came to be seen as the archetype of the "cunning or duplicitous statesman" (Oxford English Dictionary).
To illustrate the above, Christopher Marlowe's play The Jew of Malta has a prologue spoken by a character named "Machevil" or "Machiavel", who tells us, among other things,
I count religion but a childish toy,
And hold there is no sin but ignorance.
In Shakespeare's Henry VI, Part 3, Act 3, scene 2, Richard of Gloucester, who would later ascend the throne as Richard III, says,
I can (…)
set the murderous Machiavel to school.
Robert Anderson writes in Machiavelli: Bullet Guides (John Murray Press, 2012):
Although The Prince was banned under Elizabeth I and its first English language translation did not appear until 1640, versions in the original Italian as well as in French and Latin were widely read.
In An Analysis of Niccolo Machiavelli's The Prince, Riley Quinn and Ben Worthy write,
The British academic John Roe has studied the relationship between Machiavellian ideas and popular drama. He believes the Papal ban on The Prince "only guaranteed its popularity since it appeared … in a Latin translation" shortly after. It was then reprinted three times before 1600.
(…) One printer "brought out an Italian edition of [The Prince] in London in 1584, the title page claiming Palermo as the place of publication … Such audacity attests to the demand for the book." (…) Roe argues that while critics despised The Prince, the general public was excited about it.
(The first English translation of The Prince was produced by Edward Dacres.)
None of these sources appear to say until what year The Prince was prohibited. In his introduction to Robert Bolt's A Man for All Seasons (Macmillan, 1985), Leonard Smith writes,
Written in Italy in 1513, The Prince was on the proscribed list in England until well into the seventeenth century, (…)
however, without giving an actual date. This makes it still plausible that Edward Dacres's translation was published surreptitiously or using a trick such as the Italian edition published in London in 1584.
The introduction to Tim Parks's translation of The Prince (Penguin, 2009) tells us that
Sir Francis Bacon had certainly read The Prince before its first legal publication in English in 1640, defending the Florentine in the Advancement of Learning (1605) with the remark: 'We are much beholden to Machiavel and others, that write what men do and not what they ought to do.'
This seems to narrow down the end of the prohibition to a year between queen Elizabeth I's death in 1603 and the publication of Dacres's translation in 1640.
Is it possible to say when exactly the prohibition on The Prince in England ended?