HDR does effect shutter lag and that’s due to the reason of how HDR works:
If you tap the shutter button, your phone doesn’t take just one photo, it takes eight photos in a rapid row. Each of them differently exposed from strong overexposire to strong underexposure. Those eight images are combined into one be are processed in a way so that the underexposed images retain details in bright areas like clouds and the overexposed images will brighten up otherwise dark areas.
Now while the phone does shot these eight picture extremely fast, the image series is still prone to movement, both from the phone but as well as the scenery of your subject, for example if you take a picture of people dancing or take a picture out of the window from a moving car. If lighting conditions are worse, especially indoors or after sunset, you’re nore likely to encounter blurry images, because the exposure times are longer and the phone needs more time taking the required row of images.
So indoors, at night or when taking moving objects, it’s typically better to just disable HDR. Regarding XDR: I have to admit that I don’t really get how it works, it seems like an improved HDR method, but I don’t know if it either takes more shots (like ten) or if it just gets more processing time in order to get a slightly better HDR quality.