Good question.
One of the terms of Charles II's 1660 Restoration, as negotiated largely by George Monck, was the Act of Oblivion, which, in short, indicated that Charles II was not to seek vengeance for any of the injustices (real or percieved) inflicted on him, his family, or his followers during the crises of the middle decades. England was to be forgiven for any acts against the Crown and the House of Stuart in favour of future peace and stability.
Noteworthily, this clause contained one major exception - the foregiveness was not to be extended to any of the regicides: the people who oversaw the trial and execution of Charles I.
The Regicide was a controversal topic in England. The trial itself was something of a farce. Thomas Fairfax, one of the great Parliamentarian military heroes, actually resigned his commission at the head of the trial once he saw that the King's fate had been decided before the proceedings.
(Fairfax did not face recriminations - in fact his daughter married one of the King's best friends and closest companions. Another exception, proving the rule, is that Milton, who was loathed by Royalists for his vitriolic defenses of the Regicide and (from some perspectives) libels of Charles I was also spared because he didn't directly participate in the legal proceedings, though they made him sweat it out a bit.)
On top of that, the Royalist party managed to pull off one of the most successful pieces of English-language propaganda ever, Eikon Basilike, ostensibly the memoirs and confessions of the lately martyred king. It was a hyperbolic hit on the press, achieving a semi-divine status. Many Royalists started to treat the physical text with a psalter-like reverence - carrying it close to the heart, kissing it, &c. It really became a symbol for England - David Norbrook (Writing the English Republic: Poetry, Rhetoric and Politics, 1627-1660 (Cambridge: Cambridge University Press, 1999), 192.) puts it thus: via Eikon Basilike, “The king became, for admirers, and for many later literary critics, an emblem of a dying order, a lost union of church, state and aesthetic beauty which had been disrupted by a brutal modern impiety and philistinism.”
So back to the question, popular opinion in England had shifted far enough against the Interregnum establishments (Pepys has some great passages on this near the beginning of his diary, as does Edmund Ludlow) that it's perhaps fair to say that "The Nation" (so far as we can anthropomorphise it) was ashamed of the Regicide. There was little resistance or opposition to the posthumous executions of persons such as Cromwell (he wasn't the only one - Bradshaw suffered the same grizzly, weird fate). Many of the surviving Regicides were executed for treason. Others, such as Ludlow, fled into exile and died there.
Side note on posthumous execution - remember that these were people who had been buried in state. The dismemberment and such was a ceremonial part of removing them from the places of honour assigned to them, otherwise reserved for royalty national heroes.
As for Cromwell's reputation today, we'll say it's mixed. Some people romanticize his proto-Liberal stance against tyranny, but between his abuses of power - cutting England from a full Parliamentary model into a "Rump" Parliament of yes-men - and his religious persecutions, he doesn't have many big fans. He committed genocide in Ireland and was a destructive force in Scotland. The political wave which he rode to power most likely smothered Charles I's aspirations to a centralized absolute monarchy and (eventually) preserved "The Ancient Constitution," but he also led the nation down a road of blood and atrocity.