Your texture atlas has your unscaled images, so no quality is lost there. At runtime in UE, whenever the images are rotated/scaled/deformed filtering is applied by UE to map an image pixel to a screen pixel. That is happening pretty much all the time when the images are animated. Your UE camera settings can also cause filtering, since the camera determines the size of the images on screen. Generally the problem you are having is not to do with Spine, it is specific to UE and will be improved by adjusting UE settings.
I'm no UE expert so you may get better help on UE forums, otherwise my colleague Mario may have some assistance for you on Monday. I can give some general advice though.
If the size of your images (technically your atlas regions) is very different from the size they are displayed on screen, the filtered results often look bad. You may be able to mitigate this by enabling mipmapping. With that UE will use 33% more video memory to store your images scaled down by half many times. When drawing it then uses the mipmap image size that is closest to the screen size for filtering.
There are multiple kinds of filtering. Generally you want bilinear (trilinear if using mipmapping) for a smoother, less pixelated look, as it averages surrounding pixels values. Bicubic is usually only marginally better than bilinear.
I suggest looking through UE settings for mipmapping and bilinear filtering, or doing searches with those keywords.