World Turned Upside Down is a great book, and Christopher Hill was certainly one of the best historians of the twentieth century, but the work does present a distorted picture of England. Recall that the work is almost entirely about radicals, hence the subtitle, Radical Ideas During the English Revolution. The people Hill is interested in are intentionally revolutionaries or radical egalitarians - the whole point of the book is to show that the Civil & regicide opened up political possibilities never before considered in England. If you can kill a divinely anointed ruler, turning him from "His Majesty, Charles I, King of England, Scotland, Wales, and France, Lord of Ireland and Defender of the Faith" into "Charles Stuart, that man of blood" what can't you do?! It's worth stating at the outset that a super-majority of people in England, Scotland and Wales thought monarchy was the best or most appropriate form of government. To even put the king on trial required an armed coup where soldiers stood at the doors of parliament to arrest MPs and cajole others to stay away. Parliament was already down due to the Civil War but Pride's purge reduced it to a fraction of its full strength. And the number of MPs and Lords who would attend and convict Charles was smaller still.

Returning to the outline of your question, the answer is complicated. Different Stuart kings and queens had varying degrees of popularity, and their popularity fluctuated during their reign. James I had an uphill battle in 1603, being both a foreign monarch (as King of Scotland) and a man whose protestant credentials were still unproven. James' popularity was never outrageously high, but his reign was largely peaceful and prosperous. The mourning at his death was genuine, and though he was no Gloriana (Elizabeth - and even her popularity was fading by the end) he was a monarch who the people thought ruled fairly, reduced religious discord, and for the most part, kept the nation from war. There were those who complained that he spent his money too freely, kept favorites to the detriment of other nobles, or that he had illicit sex with men, but these were a minority opinion. No one, save Guy Fawkes, thought killing James or rebelling against him was a sound idea. James got a bad reputation, historically speaking, because of the writings of Anthony Weldon, who has fallen out of favor as a source in recent years (and it's been disputed if Anthony Weldon actually wrote the slander normally attributed to him). The Court and Character of King James published in 1650 destroyed James' reputation for much of the succeeding centuries.

Charles I was more of a mixed bag. His elder brother Henry (d. 1612) had been incredibly popular with the common people. He was considered handsome, athletic, and chivalrous - a good combination when considering the masculine ideas of kingship. When he died in 1612, Charles I was a boy of 12. Still recovering from the rickets of his youth, and had not been groomed for kingship the way Henry had. Charles had a stutter, was more reclusive to the point of being thought aloof, hunted less than Henry, was more interested in art and religion, than war and jousting. Though he had 13 years to be prepared for the throne, he never gained the popularity his brother or father had. On his ascent to the throne, he was reasonably well liked, perhaps in large part due to his campaign for a war against Spain (a popular cause among the Catholic-hating commoners). Charles was a somewhat successful monarch for the first few years of his reign. His reputation began to suffer as he started imposing unpopular policies. The forced loan of 1626, the collection of ship money in the 1630s, the implementation of a more ceremonial style of church worship, the separation of parishioners from the Eucharist by rails, replacing communion tables with altars, and so on. One reason for James' success was his intentional suppression of religious dissidence. Preachers were banned from discussing controversial subjects, be it double predestination or the regulation of "things indifferent". Charles took a different view, and instead he and his Archbishop William Laud began implementing a more ornamental style of church worship, and instead of banning debates, he instead forbade dissent. The difference here is that James' laws affected those who wanted more radical Protestantism and those who wanted more ornamental styles of worship; Charles' restrictions were most felt by Puritans and people more sympathetic to Presbyterianism. After Charles dissolved the parliament of 1629, he refused to call another one for a decade. This would have been more acceptable if he didn't try to extract money from the populace (most famously through ship money) without parliamentary approval during the intervening years. By the time of the calling of the “Short Parliament” in 1640, there was a significant backlog of complaints that the MPs felt needed to be redressed. Perhaps all this could have passed, but Charles had made a greater blunder in 1637 with the attempted implementation of the Scottish Prayer Book.