Well, I have no idea how to fake distortion on a UI material, but if you want to try to cheese it, maybe you could just render target something with the original material on it?
You can distort portions of a texture by manipulating its UVs with a normal map. I’ve attached an example material below. You can also see the flames mask I extracted from your background. In your UMG UI you can have the flames be a separate layer (an Image component) on top of the background, so only the flames will be distorted AND you can make a material instance for each one.
How you mix the distorted flame with your background is up to you, either in the material or having the flames be their own Image in your UI with the distorted material. In my example I’ve done it in the material to make it easy on myself but I would suggest going the other route.