Of course there were alternative churches.
For example the Jews were emphatically not part of the established church. Their synagogues were built by them, and their property was not dependent on land held in benefit permanently from the king and enjoying his protection.
Perhaps Henry didn’t think that the Established Church had kept their reciprocal obligation. He wanted a son, and the Spanish Catholic princess to whom he was married was unable to give him one. In return for his benefits, he perhaps expected a certain flexibility, and when he didn’t get it, he felt his need to keep the reciprocal obligations somewhat lessened.
Not Christian ones . . . though there were sects, Lollards, etc. And we’re discussing England, not the Orthodox in the East. Jews have always been persecuted, I dare say far more in Protestant countries than Catholic (though neither is without guilt).