Homemade double/heavy cream separates after setting in the fridge
You cannot make double cream by mixing milk and butter, this is wishful thinking. Some recipes which call for heavy cream, e.g. a cream soup, can indeed be made by adding milk and butter in such quantities that the fat ratio becomes similar to double cream, and will be successful. But for all such cases, there is no reason to mix the milk and butter before adding them to the other ingredients, you can step the step which supposedly makes cream. 

A mixture of butter and milk will always be unstable, no matter the temperature at which you added them. You could in theory avoid it by using an emulsifier and a very thorough mixing method (high powered blenders are good here), but the result will have a different taste and somewhat different texture from cream and still won't whip. I've never seen a use case where doing it makes any sense.