The other thing to keep in mind is Henry VIII didn’t forsake the Catholic Church because he disagreed with their teachings, because he believed a King should be able to divorce at will, or even because of any real disagreement with most of mainstream Catholicism.
Henry kicked the Catholic Church out of England because he essentially would have no mortal on this earth sit above him or hold anything over him, and since the Pope was doing just that with the legitimacy of his marriage Henry decided he and England did not need the Pope.
Many of Henry’s courtiers were more firebrand/radical Protestants, and thought that Henry’s actions were going to bring about a true Protestant revolution in England. Henry even dissolved the monasteries (a move probably done because it enriched him immensely.) However, if you follow the life of Henry VIII you’ll notice that pretty much the Church he left England with was very similar to the Church it had started with. Church of England’s rituals, sacraments, customs and et cetera were extremely similar to Catholicism of that era.
Very specifically note that nothing in the Anglican Church was significantly different from the Catholic Church in regard to marriage. Just like the Catholic Church of the time, Henry’s Anglican Church did not recognize divorce and remarriage as being legitimate.
However, just like in the Catholic Church there were ways around this. In the Catholic Church divorce and remarriage were not legitimate, but if say, the Pope or one of legates agreed with your argument he could grant a special dispensation. However what you had was still typically not called a divorce, but a special type of annulment.
Just like when Henry married Catherine, an act that went against proper Catholic doctrine, the Pope could grant a dispensation if the circumstances were right. In reality of course it was all political, the Popes of that age were immensely corrupt and dispensations for such things would be granted based on geopolitical concerns. In Henry’s case, his wife had a very powerful family, and a family far closer (physically) to Rome than Henry was, and with powerful armies whose wrath the Pope did not wish to incur. There wasn’t anything especially ludicrous about Henry VIII’s desire to have his marriage to Catherine annulled. Was his technical argument ludicrous and fabricated? Of course, but Kings had done such things in the past and compliant Popes had gone along with them in the past, but in this situation other pressures on the Pope were great enough to prevent them from agreeing to the annulment.
When Henry finally rejected the papacy he did not just simply declare that his marriage to Catherine was over. He went through the channels of his new Church, of course, if his Archbishop hadn’t gone along with it he would have been executed, but for appearances sake Henry didn’t just divorce Catherine he had a lawful and appropriate annulment under the doctrines of the Anglican Church.
Anne Boleyn was executed for treason, and there was no special prohibition about an English King marrying again after being widowed. Also due to Anne Boleyn’s treason conviction I imagine (although don’t know for sure) the Anglican Church probably officially declared the marriage with Anne unlawful and void at some point and possibly ascribed it to Anne using witch craft on the King (I even believe such an argument is what initially disinherited Elizabeth.)
Jane died of natural causes and thus there was not any special dispensation required.
Catherine Howard died in the same way as Anne and following a conviction for treason, so it was dispensed in the same way.
Anne of Kleves and Henry had an annulment. Again, it went through the official and appropriate channels of the Church of England, it wasn’t just a divorce. Officially Henry and Anne never consummated their marriage and it was not valid. In this case there is even some evidence Henry and Anne of Kleves genuinely never consummated the marriage, surprising a notoriously promiscuous King like Henry would be particularly picky.
Catherine Parr of course outlived Henry, and she even married after Henry died. She was even married before, but in her case she was a widow when she married the King.
So as you can see there was nothing particularly divorce friendly about Henry’s new church. Henry didn’t seem to believe you should be able to just divorce on whim, and that’s certainly not how he interpreted scripture. He mostly agreed with the Catholic stance on divorce. He however was very good at convincing himself of the righteousness of his own situations. While opposed to divorce as a concept, he was good at convincing himself of things like his first marriage being immoral and invalid because his wife and brother actually consummated the wedding, thus making the Papal dispensation invalid. (It’s highly unlikely that happened.) With the rest of his wives that he separated from, it was following either their conviction of treasonous acts and subsequent executions, so dispensing the marriage was not a particularly difficult issue. With Anne of Kleves there was a genuine situation that probably would have been annulled without fuss by the Pope of the time, if Henry had still been in the Catholic fold.