There is probably a more direct link between those who DON'T believe in religion and those who DON'T support the (so-called) Monarchy.
Societies across the world have, for centuries, tended to follow orthodoxies, rather than to challenge them. For example, if my parents were asked to enter their religion onto a form, they'd almost certainly have filled in 'Church of England'. That wasn't because either of them actually believed in God but simply because, being British, that's what they'd been brought up to do.
Similarly, people from working class backgrounds almost universally voted Labour ("because that's what our family have always done"), whereas those from the more privileged classes automatically voted Conservative ("because that's the party for people like us"). Equally, people supported the monarchy, simply because that's what they'd been brought up to do, rather than because they'd actually given any thought to why one group of people should be regarded as 'superior' to others simply because of who their parents are.
These days, fortunately, schools teach young people to constantly challenge orthodoxies. As a result, many people grow up actually examining the policies of political parties before deciding how to vote. They also think about whether Christianity, or any other religion, makes sense before deciding whether or not to believe in it and whether there's any real point in the Monarchy before deciding whether or not to support it.
So those (often older) people who're happy to leave orthodoxies unchallenged tend to believe in God (or, at least, to say that they're 'C of E', even when they don't actually believe in a deity) and, equally, they tend to support the Monarchy. Meanwhile those who've learned to think for themselves tend to reject religion as being preposterous and the Monarchy as being an outdated anachronism.