The late Queen Elizabeth IIās coronation speech in 1953 included one key line, in which she promised to be āDefender of the Faith upholding the rights of the Church of Englandā. This is a title which has been used as part of the full style of many English, Scottish, and later British monarchs since the early 16th century.
Well, I doubt very much that Scottish monarchs gave much of a
tit about the CofE, frankly. After the Union of the Crowns, James VI & I and Charles I only liked it because it was weedy enough to give in to their ideas about the divine right of kings.
The Scots have always been stroppier about their religion, whether it's hurling stools at pulpits, fighting civil wars and uprisings about religion, and willing to die for Presbyterianism in the case of the Covenanters. They made Charles II swear an oath to uphold it at his coronation at Scone and even now the monarch has to sign an agreement to uphold the Church of Scotland, despite it not being an established church. The stroppiness of the CofS extends to its governing body, the General Assembly, which didn't hesitate to lay into UK politicians in lieu of a Scottish Parliament. It's also quite forward thinking too, having had women ministers for years before the CofE, most being quite happy with gay ministers and equal marriage, and seeing no problem with divorce, hence Anne and Tim getting married at Crathie Kirk unlike Charles and Camilla who could only get a blessing of their civil marriage.
Apart from various Northern Irish descended bigoted lunatics, no one really gives much about of a
tit either about a multitude of religions and churches. Eh, you do you, pal; it's between you and your god, the basis of our version of Calvinism and the reason for the insistence on education. You can have a legal pagan handfasting too as well as celebrating many other religions.
I really can't think of why Charles shouldn't change the coronation oath; it's not as though many people are actively Christian, let alone a majority of the population.