I would be more worried about younger Henry being perceived as a threat? Was he still being considered for the church at this time. I would still think that even with the death of Arthur that Henry VII would make Henry some noble title as he did in OTL. the whole, 'heir and a spare' rule would still apply.
Henry VIII as a young man was a cultured and civilised man who held himself to a very high moral standard compared to many of his European contemporaries, his only acknowledged bastard the Duke of Richmond wasn't born until 1519. He was no Richard III, he would have respected the succession and would have been his nephew's most loyal servant. Even after Henry's attitudes became more extreme in the late 1520s and early 1530s, he was still deeply opposed to some proposals made by his ministers that he declare his bastard the Duke of Richmond his heir, that was not legitimate in Henry's eyes.
It is still possible that the plans for Henry to enter into the Church would have been delayed or abandoned with Arthur's death. I don't think Henry would have himself embraced being a Church man and I think with his father's death in 1509, he would have taken over the regency for his young nephew and been the power behind the throne.
Had Arthur, Prince of Wales died and Catherine is pregnant and produces a male child after his death, the child is the undisputed heir to the throne after his grandfather King Henry VII.
Catherine's pregnancy and the birth of a healty male child has huge significances for Catherine. She is no longer the widowed childless spouse of the former heir, now she is the mother of the future monarch. She will be the second most important woman at court after the Queen, Elizabeth of York.
I suspect Catherine will embrace motherhood as she did with her own daughter Mary Tudor, and she will likely not spend too much time at Court but will spend her time mostly in the country with her son (to ensure his health) in palatial luxury. She will be happy and content, I suspect she will not fall into the trap of many widowed royal ladies and will avoid romance, she was too high born and principled.
Catherine may venture more into politics as suits her and of course her child will be raised to be a good Catholic and to have a particular alligance towards the Empire of Charles V, Catherine's nephew.
It occurs to me that Catherine being delivered of a healthy male child has consequences for the wider Tudor dynasty.
It is written that in the immediate aftermath of Arthur's death, his mother Elizabeth of York tried to comfort her husband by pointing out that they still had a living strapping son and two healthy daughters and were both young enough to have more (a quote attributed to Elizabeth which I have always thought dubious but for these purposes we will run with it). Elizabeth proceeded to fall pregnant after the death of her first child and duly died shortly after giving birth to a female child who also died in 1503.
Elizabeth's death is traditionally identified as being disastouros for Catherine of Aragon as Elizabeth was seen as a sweet and caring woman
and would not have tolerated her husband treating Catherine the way that he did.
It is entirely possible that if Elizabeth became a grandmother to a future King of England in the Summer or early Autumn of 1502, there is a possibility that she may not have herself become pregnant (her last pregnacy had been more than 4 years previous), thus preventing her premature death. Elizabeth was 37 at the time of her death, not old by modern standards but by 16th century standards it was risky to have a child at any age, let alone when nearing 40.
I suspect that Mary Tudor the elder will end up marrying Charles V rather than the elderly Louis XII. It creates the possibility of much stronger ties between the English monarchy and continental Europe.