How to thicken a Yoghurt based cold sauce?
The usual way is to use a thickener. Some of them require warming, but others do not. Guar or xanthan gum will work if used in the cold sauce. It is the easiest way. If you don't have them, yuo can use gelatine, but you'll have to dissolve it in warm liquid first and then add to the cold sauce, then wait to thicken. None of these will change the taste. 

If you are from the "no additives" fraction, you can just use a thicker dairy product. I don't think the heavy whipping cream would be a particularly good fit to yogurt. The best choice would be a fermented product thicker than yogurt, e.g. sour cream or creme fraiche. But you can also use a cheese, although this will change the taste (without making it bad or too different from the original). Good choices would be cream cheese, ricotta, quark, tvorog or mascarpone.