I will discount the 3rd and 6th wives, Jane Seymour and Catherine Parr, because any dissatisfactions Henry had with these 2 wives did not stem from their inability to bear him male heirs.
Henry’s 3rd wife Jane Seymour gave birth to his male heir ( who succeeded him as Edward VI) less than 18 months after their marriage. She died soon after giving birth.
By the time of his 6th marriage to Catherine Parr Henry was struggling with morbid obesity and illness and was probably completely impotent. Even if Henry had been up to the job, he had no high expectations of having children with Parr - she had been widowed twice previously without ever having been pregnant and was presumed to be infertile.
But as to Henry’s other 4 wives - why does it seem that he was patient with Katherine of Aragon (K of A) and not with the others?
Firstly age. Henry was 17 and 1st wife K of A was 23 when they married. Like all kings of the period, Henry wanted male heirs quickly, but as a newly married 17 year old he felt that he had plenty of years for that to happen.
Importantly, K of A was not infertile in her younger years. She conceived fairly easily and had had 6 pregnancies in 9 years. Only 1 child survived longer than a couple of months(future queen Mary Tudor) But 3 of the births/ stillbirths were boys who could have become Henry’s heirs. One boy born to the couple (named Henry for his father) lived for 2 months. Put simply, for many years Henry had hope that it was only a matter of time before his queen would give birth to a healthy boy.
Important to note that Henry did not see K of A’s value to him as residing only in her ability to be the childbarer of his heirs. Unlike his subsequent queens, K of A had been raised and educated to fulfil the role of queen. She was an informal advisor to Henry, she had high status, and powerful connections and relatives in Europe. Henry trusted her, at one point, to take the role of regent while he was out of the country.
It should be noted that although Henry was married to K of A for 23 years he wasn’t patient about the lack of male heirs for all of that time. As Henry and K of A became less youthful, and she was no longer getting pregnant, Henry began to have concerns. He fell for Anne Boleyn and began the process of trying to extricate himself from his 1st marriage. It was a lengthy battle taking 8 or nine years years to complete. If K of A would have quietly agreed to an annulment of her marriage and to fade into the background when Henry first raised the idea, her marriage to Henry would have been of significantly shorter duration.
Henry might be said to have had minimal patience with wife 2 Anne Boleyn. Anne Boleyn talked a big game throughout her courtship period with Henry (while he was still married to C of A) and in the early part of her marriage. If she had any private doubts on the matter of whether she would bear a male heir, she didn’t voice them. Henry was deeply disappointed about the sex when their child (later Elizabeth I) was born in the first year of their marriage, but he was still in love with Anne and was happy for them to try again. Two miscarried pregnancies later and Henry was on red alert. He and Anne were not in their teens or 20’s. He wanted a male heir and he didn’t have available the number of years to keep trying that he’d allowed to K of A. And if Henry had, rightly, seen Queen Katherine (even without a son) as enhancing his prestige, Queen Anne without a son brought no such advantages to the table.
Henry’s marriage to foreign noblewoman Anne of Cleves was unconsumated and lasted only 6 months. Henry knew very quickly after his wedding that he had neither desire or ability to consummate his 4th marriage and so there would be no possibility of male heirs. His only wish,from the very day of the wedding onwards, was to exit the marriage as fast as possible.
Middle aged Henry was besotted with his teenage bride 5th wife Catherine Howard. Despite the huge difference in their ages, his ill health, and that he was probably mostly impotent, Henry expressed high hopes that Catherine would get pregnant. It didn’t happen during their short marriage of less than 2 years. But Henry ended their marriage because Catherine was convicted of adultery, not because he was annoyed that she had not conceived another male heir. Would Henry have started, at some point, to grow angry about Catherine’s lack of pregnancy? Possibly, but there’s no way of knowing for sure, because their marriage never reached that stage.
Most people have already excellently answered key reasons. Katherine's relatives and significance definitely would still have saved her if she faced the same charges for Anne Boleyn and Katherine Howard.
But as has been implied in other comments. Henry was only desperate for a male heir for his first three wives. And even in this area, it is a little more complicated than people understand. His last three wives he had vaguely hoped to get "a spare" but Jane's Prince Edward had resolved the issue of succession for Henry.
As stated before, Henry and Catherine of Aragon were married for a long time, because they married when they were both young and Catherine had given Henry a son one year into their marriage. Unfortunately the baby died. Catherine tragically would go through her reproductive years unable to give Henry the son he needed. This issue only became a pressing as Henry was in his 30s and Catherine went through menopause.
As Henry began to worry about the succession, in came Anne Boleyn, who knowing from her sister's experience the life of a King's mistress, resisted Henry's advances (not the least because Henry had ruined her earlier engagement). But and this is somewhat divisive. Per an argument of Alison Weir in The Six Wives of Henry VIII, Henry would have divorced Catherine regardless of Anne. He needed a male heir, and Catherine had not given him one and could not. Henry also, incredibly devout and romantic (in the Medieval sense) looked to Scripture to "explain" not having a son with Catherine, as his mistress Bessie Blount gave him a son, Henry Fitzroy.
Now, to be clear, Henry leaving Catherine was incredibly hard for him, politically, religiously and personally. Personally because Henry genuinely loved Catherine. I personally think she was the only wife Henry genuinely loved as he took her counsel politically, permitted her to be his regent in his absence and even though she was a powerful ally she did not give Henry any dowry and the Anglo-Spanish alliance was not greatly strong. In contrast he was almost always volatile with Anne, and tersely warned Jane never to speak to him about political matters, was repulsed by Anne of Cleves, spoiled Katherine Howard, and nearly killed Catherine Parr for disagreeing with him. Although he did not want to be married to Catherine, mostly for succession (and religious) reasons he always cared for her. Catherine's death and her farewell letter to Henry deeply saddened Henry.
For a while (and possibly until his untimely death) Henry always had his bastard son in mind as his "spare" heir as he granted Fitzroy the Dukedom of his Father. Fitzroy was the most powerful peer in the realm next to the King. But a recognized bastard was always a shaky heir compared with a legitimate son.
So, with Anne, and in a completely unqualified psychological evaluation. Henry was less in love and more in passionate lust with her. She became the ultimate sexual conquest for him. When he finally had Anne he came to recognize that she was not his ideal for a Queen as she was argumentative, passionate and confrontational. In Henry's POV after such a long and hard effort to be with Anne, for her to not give him the son he needed and being more of a pain was galling.
But, getting at a bigger question why did Henry want Anne dead? That is also a complicated question. Now part of this is tied to how much is Henry the puppetmaster or puppet debate.
Essentially because Anne had not given him a son, because Catherine was dead and no longer the only woman he could realistically marry other than Anne, and because Henry had fallen in love with Jane. Henry wanted to be rid of Anne, so he turned to Cromwell to give him a reason.
Cromwell had seen how Henry's previous no. 2, Cardinal Wolsey, was removed from power and left to the mercy of his enemies. Cromwell concerned for his own livelihood found every morsel of innuendo against Anne and tortured "confessions" out of the peasant "lovers". While this was grounds for divorce, what may have actually led to Anne being beheaded was Cromwell taking some words from Anne that were harmless courtly flirtation at face value that sounded like Anne was plotting to kill Henry. That is what led to Henry being willing to kill Anne.
So, fast forwarding to Katherine Howard. She was guilty of adultery. But what may have contributed more to her death was that she was by religious traditions at the time married to another man, making her marriage to Henry polygamous which would be horrifying and infuriating to devout Henry, leading to punish her less for adultery and more as a sinner, but akin to Anne he was "merciful" and had her beheaded rather than burned.
I think you have a distorted sense of the timespan and situation from the many fictional depictions of the period. As I discussed in my previous answer, Why did Henry VIII marry “old” wives? ...
... Henry believed that if he were with the right woman and God blessed their union, a son would be granted to them. Remember that a major factor in his stated reasoning for divorcing Catherine of Aragon was that he claimed to believe she and his brother had consummated their marriage, and that therefore he had sinned by marrying her and was being punished with daughters, stillbirths, and children who died young. If he corrected his error, then God would be like, "Now we're good!" and provide a healthy, male heir to the throne.
When Henry met Anne, she was about 25 (and he was about 35), and he was highly attracted to her. She initially rejected his advances on the grounds that good girls don't get involved in extramarital/premarital relationships, which led to his desire to make her his mistress morphing into a desire to make her his wife, and despite the fact that it took seven years, he followed through. Catherine's track record of unsuccessful pregnancies and apparent menopause were a large factor in his interest in a younger bride, but by all accounts Henry was in love with Anne. His feelings were strong enough to keep the unconsummated relationship going for years and to result in a massive break from the Catholic Church and from an alliance with Spain and the Holy Roman Empire. And at the same time, 32 is not old for childbearing - certainly it would have been an unusual age for a first birth, but women typically gave birth into their thirties. Given that Anne became pregnant pretty much right away after the marriage, it seemed reasonable to believe that the problem was fixed.
By the time Jane came around, Henry had had time to reconsider. The theory had been that if Henry ended his marriage to "his brother's wife" and took up with someone not related to him, he would get a son. Instead, after he married Anne, they had a daughter and three miscarriages. Was God still disapproving? Henry was attracted to other women, as he had been in his previous marriage, and set his sights on Jane Seymour, who was one of Anne's ladies in waiting, not related to him in any way, and not associated with Anne and the break with the Church. She was also well within her childbearing window, as evidenced by the fact that she gave birth a year after the marriage to a son who was healthy enough to survive his childhood. All of these factors were important. ... In any event, Henry's reasoning was that he didn't need to have a wife who was at peak fertility or young enough to allow for twenty years of trying for an heir: she just needed to be able to give birth once or twice, and if the marriage was sound according to God those births would be male.
Once he had a son, there was much less pressure to find a wife who would give him another, and his reasoning for his treatment of the latter three wives had nothing to do with their ability to do so. Anne of Cleves, who was married to form a Protestant alliance, never slept with him by his own choice; Catherine Howard was a spectacularly unorthodox midlife crisis half-his-age wife; and there doesn't seem to have been much to-do over whether Catherine Parr conceived. (Given that she'd been married twice before and had no children, it seems unlikely that he would have picked her for that purpose, rather than her familiarity with his family.)
His first wife was the youngest daughter of Isabelle of Castile and Ferdinand of Aragon. These monarchs had funded Columbus's expeditions to the New World, expelled Muslims from Spain, etc. Her sister had married into the Hapsburg dynasty that would dominate much of Europe. Her relatives also maintained a relationship with the Papacy. Cutting off her head could've led to a war with quite a few European powers.