Moderates don't ever tend to do too well (see French and Russian Revolutions) so I think it was fairly inevitable that it would either fall into a more Protestant stance (as it did OTL) or lapse back into Catholicism eventually.
That said, if it had remained "Catholic without the Pope", I think there are a few major changes which could occur. First of all is in Ireland, where ITTL Anglicanism will do a lot better at replacing Catholicism and perhaps even replace it altogether as this will be a pill much easier for the Irish to swallow than was historically the case. After all, the Irish Church will only be replacing one foreign leader with another, with most of the doctrine and traditions staying the same. This won't stop the Irish hating their conquerors, but may allow the English to better colonise Ireland (as the religious aspect will be removed/reduced) so we may see a more English-settled Ulster and perhaps even a resurgence of the Pale. If Ireland becomes independent ITTL, the Church will not be so important to them (as it will not have become intertwined as an integral part of their national identity) and will therefore play a smaller role in shaping the politics of an independent Irish state.
Secondly, such a situation in England may well lead to similar situations in other European countries as other rulers follow Henry's example. Perhaps they play a significant part in the Reformation by providing a "third way" between Catholicism and Protestantism, or maybe TTL's Counter-Reformation isn't led by Rome but rather by local Catholic rulers who take matters into their own hands to reform the Church. We could easily end up with a Catholic Church who's rituals and doctrines are familiar but the organisation of which is almost completely different from OTL, with different local Churches and far less of a central authority. This alternate Gallicanism (probably called Albionism or something instead) would become the most common practice in the Church, with Ultramontanism being viewed as archaic and only being found in a few small and deeply traditional countries by the Late Modern Period.