Did the church continue to prosecute scientists for heliocentrism? If not, why not? What changed?
Why didn't the church go after Isaac Newton?
This will answer two out of three parts of the question:
(a) 'Why didn't the church go after Isaac Newton?' It was not at all the whole church that was involved in the Galileo affair: it was the establishment of the Roman Catholic church of the time. In much of (mostly northern) Europe, the Roman Catholic church had no authority at all: the reformed churches predominant there did not acknowledge the Pope (and usually disagreed with what was done in his name). The church of England was one of the reformed churches, roughly speaking, and was generally anti-Pope, so that Newton was in no danger of suffering the same thing as was done in the Galileo affair. Evidence to show how the reformed churches were completely independent of Rome and generally opposed to what the Pope did can be seen, for example, in sources that show how the reformed churches even refused for a long time to accept the (Roman, papal) Gregorian reform of the calendar: see M Hoskin, 'The reception of the calendar by other churches', pp.255-264 in 'Gregorian Reform of the Calendar', (conference proceedings ed. G V Coyne et al., 1983).
I hope others with better familiarity than mine with the Galileo affair will answer the part of the question referring to 'prosecution for heliocentrism', because from what I recall of careful historical accounts I believe it can be questioned whether what happened was simply a 'prosecution for heliocentrism'.
(b) 'Did the church continue to prosecute scientists for heliocentrism?' In any event, whatever the details of the Galileo affair, there was little or no obstacle for those under Roman Catholic authority who wished to discuss heliocentrism or Newtonianism. Evidence of that can be seen in both the existence and the content of a well-known annotated edition of Newton's 'Principia' produced in 1740-42 by two Roman Catholic priests who were also teachers of mathematics: Fathers Thomas Le Seur and Francis Jacquier (members of the 'Minim' monastic order, but their edition was sometimes mistakenly referred to as the 'Jesuit edition'). At the start of the third book of the Principia they inserted a declaration, stating in effect that it was not possible to discuss the book without speaking with Newton's voice. For their own part they accepted the papal decrees against the motion of the earth. And that was it: it can be seen from the content of the book that after making their brief statement they just carried on giving the text and discussing it in the notes as if there was nothing against it. (Their declaration can be seen here, along with the rest of the volume, in a digital scan of an 1822 reprint of their edition -- their edition was valued for a long time for its annotations and commentaries: https://archive.org/stream/philosophiaenatu03newtuoft#page/n16/mode/1up).
"The church", as you call it, had nothing to say in Protestant England.
This question discusses the common assumption that the only issue of the Galileo trial was heliocentrism. Briefly, some scholars have argued that an additional issue was atomism and its difficult relation with doctrinal issues. Even as far as heliocentrism is concerned, the opposition was not to using it as a technical hypothesis in scientific calculation so much as the more sweeping assumption that it describes the "true" state of affairs.
What changed? Less and less people believed in the church. So the church lost its power. (Otherwise they would certainly have killed Darwin with delight.)
It got...difficult. At one point, there was a model where all the planets except Earth revolved around the Sun, but Earth revolved around it's own sun...
In all honesty, some people still don't believe it.
EDIT: re: Newton ... If you read principia Newton goes out of his way to avoid religion. As a weird historical sidenote, Newton knew Darwin's father, through the church.
EDIT^2: It was his grandfather Erasmus Darwin, not his father Robert Darwin.
To answer the last part of the question, Isaac Newton lived in England, and the Catholic Church had been supplanted as the country's highest religious authority by the Church of England.
The reason was that King Henry VIII wanted a divorce from his first wife, Catherine of Aragon, so that he could marry Anne Boleyn. The Pope refused him the divorce, so Henry "disestablished" the Catholic Church and set up the Church of England in its place, answerable to the king (or queen) of England.
Other "heliocentric" scientists, e.g. Kepler, lived in Germany or other parts of northern Europe where Catholicism had been replaced by Lutheranism or some other Protestant faith.