This is a great topic, Jord! Henry never spliting with Rome and England staying Catholic is my personal favorite POD. I think England would probably develop incredibly differently; I doubt it would look anything like OTL. Here's my thoughts:
1. Protestantism would definitely catch on in some parts of England; it was already catching on IOTL. I think Protestantism would be most popular in the southern counties and in London, where it was strongest IOTL. If Henry never finds anti-papalism politically convenient, the monarchy's response will probably continue to be fierce, as IOTL. In the 1520's Henry was intensely hostile to Luther and other Protestants, and if he never splits with Rome I see no reason why that would change (indeed, even after the split with Rome he sometimes cracked down pretty hard when he thought the Reformation was getting too extreme, such as in the 1540's after the downfall of Cromwell). Henry (and Lord Chancellor Thomas More, who probably won't get to be a martyr ITTL) will keep on burning heretics and inveighing against Luther in printed polemics. There might be an English Protestant exile community on the continent, like the Marian exiles, but unlike the Marian exiles these Protestants will probably never return to England. On a related note, without important Marian exiles returning after being exposed to Calvinist ideas in Geneva, the brand of Protestantism prevalent in England will probably continue to be much more heavily Lutheran-influenced.
2. Henry staying Catholic might not necessarily save the monasteries, at least not all of them. Monastic dissolution was motivated by greed as much as religion, and Henry VIII is going to be a greedy SOB with delusions of grandeur no matter what religion he practices. Who knows, having a surviving son to pass his realms on to might make him even crazier about conquering France? As long as he doesn't have enough money (which he never did) and he feels the need to engage in foreign escapades (which he always did), at least the smaller English monasteries are exposed to reappropriation by the Crown.
However, the bigger monasteries would probably survive. I'm not sure how this affects economic development, but it would be very highly culturally significant. The present day world of this ATL would probably still have all kinds of medieval artwork and literature that was destroyed by the Protestants IOTL. This might radically alter their understanding of history: the idea of the Middle Ages as a Dark Age might never be tenable if more from that era survived.
3. I doubt this POD would affect Ireland all that much in the short term, honestly. Protestantism never really caught on there in the first century after the Reformation started; it only really arrived through the colonization of Ulster in the 17th century, which is probably butterflied away here. Which isn't to say peace is going to break out in Ireland: instead of Protestant vs. Catholic conflict, we might just see a continuation of the Old English Catholics vs. Gaelic Irish Catholics vs. New English Catholics/royal administrators conflict.