Indeed, I had forgotten about England. I'll have to think about it, because it was absent from my long list of priors. Here in the US, it's generally assumed that the countries "across the pond" are less religious.
The US is presently ruled by a overtly religious party that rejects church-state separation. Our President sells Bibles and prays in public with his cabinet. His party is supported by a predominantly religious electoral "base." And the influence of this system is not limited to one errant president, but has been systematically pursued for decades.
I hope that their ideas are distortions of true religious doctrine, but I can't prove it, and have no power to challenge them except in the voting booth.
I learned an alternate take on "give unto Caesar" which was that Caesar's money represented the wealth of this world, which is worthless compared to the infinite wealth of the spiritual world. The point was for Christians to remain aloof from earthly problems such as governance, which were expected to be temporary.
I don't doubt that religions promote virtuous ideals. And I don't expect religion to vanish, so if democracy has religious roots, those roots won't suddenly be forgotten.