The Science Museum model has an all-gilt coat of arms (and everything else) I see, but the proper colours for the arms would seem to be:
Top-left and bottom-right quarter: The French fleurs-de-lis and English lions, quartered. The fleurs de lis are top-left and bottom-right, gilt on a blue background, the English lions top-right and bottom-left, gilt on a red background.
Top-right quarter: The Scottish lion, red on a gilt background
Bottom-left quarter: The Irish harp, gilt on a blue background
The supporting unicorn should be white, with a gilt chain and neckstrap.
It's the royal coat of arms, not of the King Charles II but of his brother James Stuart, Duke of York, for whom the ship was named. Being heir to the throne he was more or less wearing a Prince of Wales hat as well as being Duke of York. Wikipedia has an illustration of his coat of arms that is useful for reference.
The three feathers that sit on top of the quarter-galleries ought to be white as well, by the way. They are "the prince of wales' feathers", another device that symbolizes the name of the ship.