Depending on their personal relationship, Henry could either feel reassured at having a 'safe' and trustable brother as a potential heir if he doesn't have a son by Catherine - and so be less keen to replace her - or even more paranoid than in OTL, at not being able to trust Edmund to keep to his policies if/when he succeeds and in fear of Edmund plotting with his enemies. If he was still angry and humiliated at not having a son, having a brother he didn't trust ready to challenge his daughter Mary once he was dead could make him even keener to get rid of Catherine quickly - especially if Edmund was friends with conservative nobles like the duke of Buckingham (executed 1521 for supposed treason) and had a poor relationship with the Boleyn family. In that case, the Boleyns could expect disgrace if Henry died and Edmund succeeded, and so use Anne to push Henry into sidelining Edmund by getting a new male heir ie a son by Anne. In reality Henry's younger sister Mary, Duchess of Suffolk (d 1533), and her husband Charles Brandon, Duke of S and Henry's boon companion, backed Catherine against the Boleyns and were suspected of ulterior motives as their son Henry (1516-34) was a potential alternative heir to the throne - and the same could occur to Edmund.
It's anyone's guess if Edmund would have an interest in serious scholarly studies and theology and so be a potential convert to Protestantism; his father Henry VII organised 'high-powered' tutors and serious academic studies for his children and we have a record of the international-linked top Dutch scholar Erasmus visiting the royal nurseries at Eltham, Kent on a trip to England in 1499 (when Edmund was an infant) and Prince Henry (ie H VIII), aged 8, showing off his Latin to him. Henry and Edmund' elder brother Arthur was a studious type rather than being keen on jousting etc like Henry, quite apart from his weak health, and their grandmother Lady Margaret Beaufort (the 'Red Queen' of Philippa Gregory's books) was very learned and a college founder. Edmund could have taken after them, and if he took after the active Henry he could have led various English expeditions to France and prestige diplomatic missions if Henry trusted him.
Given the lack of princesses of the right age available in the English-allied Habsburg dynasty in the 1510s and 1520s, and Henry's usual hostility to France, possibly Edmund (b 1499) would have been married off into the family of Emperor Charles V (also King of Spain)'s Portuguese relatives , eg to King Manuel I (b 1469, r 1495-1521)'s younger daughter Beatrice (b 1504). Charles secured her elder sister Isabella for himself - and so Edmund and Beatrice's children would have been first cousins of the future King Philip II of Spain. A conservative and Catholic Edmund would have been a good ally for Philip at the time of his marriage to Henry's daughter Mary and alliance with England in 1554 - and he or his children could have been the Catholic and Spanish candidate for the throne to succeed Mary in 1558. They would be 'legitimate' as seen in Catholic eyes unlike Elizabeth Tudor, safer and nearer to the throne than the pro-French Mary Stuart of Scotland (at this point married to King Henri II of France's heir) who was grand-daughter of Henry VIII's elder sister Margaret Tudor, and ditto compared to MT's daughter Lady Margaret Douglas.
Edmund may have been able to negotiate the perils of high court politics in the 1530s-1540s successfully without being thrown in the Tower or executed by his ultra-suspicious brother , provided he had followed the current religious policies unhesitatingly , not aroused the suspicions of Thomas Cromwell in 1531-40 or Anne Boleyn in 1533-6 too dangerously, and not got involved with the Pilgrimage of Grace in 1536 or the alleged plots of Henry's half-Plantagenet Courtenay and Montague cousins with the Habsburgs on Princess Mary's behalf in 1537-9. His potential to be a 'Richard III' as regent for Henry's heir - Mary, Elizabeth, or Edward (VI) - if Henry died suddenly would have made his position perilous, eg when Henry was knocked out in a jousting accident in Jan 1536, but if Henry trusted him and he was not interested in religion he could still be around in 1546-7.
At that point, as next adult male heir he was in the usual position to be regent when Henry died - in OTL it was Edward VI's uncle on the female side, the reformist Protestant Edward Seymour (Duke of Somerset) , who headed the regency council then ignored H's wishes by getting the rest of the council to agree to him being full 'Protector' ie regent not just first among equals. So would Edmund have been in the real life position of the conservative Howards, who Seymour tried (successfully) to get Henry to destroy politically before he died to clear the way for himself? Or could a prince fight back better than the Howards, so Edmund would have blocked the way to Seymour and secured a moderate, non-Protestant regency for Edward VI - so no 'Edwardian Reformation'? At this point, Henry VIII was still theologically Catholic on most matters, though reformist and anti-Pope, and was not aiming at a Protestant regency for his son. Assuming that Edmund was regent or co-head of the ruling council after 1547, Edmund or his presumed son would be in a good position to succeed Mary Tudor in 1558. If Edmund was Protestant, then when Edward died in 1553 he would have had to weigh his religious duty to save England from Catholic Mary against his loyalty to her as next legitimate heir - so if he put religion first he could have either taken the throne or married off his son (born 1520s?) to the replacement Queen, either Elizabeth Tudor (b 1533) or Lady Jane Grey (b 1537). So a continuing Tudor dynasty and no Stuart succession in 1603?