I'm not entirely sure how successful a King Henry IX in this PoD would be - at least, I'm pretty sure he wouldn't attain the same levels of popularity as his OTL niece. But his reign would be quite an interesting one.
Personality wise, Henry wasn't exactly a cheerful chap. He was quite dour, gloomy and bad tempered which was no doubt exacerbated by a drinking problem he developed in the 1930s - which brings us to Beryl Markham. Henry had a love affair with the Kenyan aviator Beryl Markham in the 1920s and 30s and was quite cavalier about it. He didn't seem to care that people knew and committed the cardinal sin of leaving a pretty blatant paper trail. Queen Mary stepped in and managed to both end the affair and retrieve most of the evidence (at huge personal expense). She then insisted he marry immediately in case the newspapers broke the story as they were planning.
Which is how we would get Queen Alice here and I would go out on a limb and say that she might well prove to be the monarchy's biggest asset. At least, she would boost it's popularity in the face of yet another dull King in the model of George V. In the OTL, Alice was extremely popular with the public in the 30s, 40s and 50s and when the Duke was sent to Australia as Governor-General, she drew enormous crowds whilst he mostly confined his activities to military engagements. I don't think it's any secret that their marriage was one of convenience but she was determined to make it work even though he wasn't an easy man to live with. She kept herself busy by embracing her public role despite her intense shyness and I can see the public taking to Alice in the same way perhaps they took to Queen Elizabeth, the Queen Mother (though it must be said Alice was nowhere near as calculating or difficult as Elizabeth behind the scenes) and if he's going to be successful as King in terms of public affection, the donkey work is going to fall to his Queen consort.
The 60s/70s would be the most interesting development in this WI though. Henry suffered a stroke whilst driving back from the state funeral of Winston Churchill in 1965, crashing his car and nearly killing Alice who had fallen asleep on the journey. In this POD as King and Queen, they're not driving themselves but the Duke's lifestyle undoubtedly leads to the same medical emergency. From '65 onwards, Henry was partially paralysed and couldn't speak. Alice nursed him devotedly but he wasn't seen in public ever again. In this WI, you either have a regency or an abdication with the Prince of Wales (William) stepping up to the plate. But he's got his own problems. As has been mentioned, he was diagnosed with porphyria (though we have no idea how intense his symptoms were) but perhaps most important was his relationship with Zsuzsi Starkloff.
Starkloff was a Hungarian model and former airline stewardess who started her relationship with William in the late 1960s. William was devoted to Zsuzsi and even though the wider Royal Family was opposed to them marrying on the grounds that she was a Catholic and he was expected to become a working member of the Royal Family, Henry and Alice were quite supportive and received Starkloff at Barnwell. Their advice was that the couple should wait a while until the mood changed - they last met in 1970 because it was made clear to William what he'd have to sacrifice to marry Zsuzsi and almost immediately ran into the arms of Nicole Sieff - a divorcee. The question is how these relationships impact his life when he's Prince of Wales, how Henry and Alice might react and what public opinion would be considering William's enormous popularity.
If we stick with the OTL events, in 1972, William was killed in an air crash - if the relationship with Starkloff is still ongoing, you've just given conspiracy theorists a new project - and just two years later, Henry suffered another stroke and died. In this PoD, Henry IX (if William was regent) would be succeeded by his second son who becomes King Richard IV. He's only been married to Queen Birgitte for two years here with the couple's first child (a son) born just months after the accession. Which would increase the work load on other members of the Royal Family quite a bit and this POD, you actually don't have that many working royals in the fray at all.
In 1974, you'd have Queen Alice, the Queen Mother (if she chooses to use that style), King Richard IV and Queen Birgitte, but as to other royals who can cut ribbons etc? You're only left with three; Edward, Duke of Kent, Princess Alexandra of Kent and Prince Michael of Kent. If Michael marries the same way (and it's doubtful here because Lord Mountbatten's influence is severely curtailed), he takes himself out of the running when he marries in 1978 - bringing the Royal Family to just five working members. That's not necessarily a bad thing but realistically you might expect public engagements to be cut because of a lack of manpower which in the 1970s, isn't exactly ideal when people start to ask about value for money and the continued role of the monarchy. But then, perhaps a smaller Royal Family is more advantageous in this situation?
