Is the changing pad just a single piece of fabric? Is it pieced? Will it have a solid back? When I do receiving blankets with rounded corners (2 layers thicK), I do an envelop style finish.
Here is what I do: Finish the top and quilt it to the batting (for receiving blankets I do not use batting), but not the backing fabric. Cut the backing fabric to exactly fit the top and batting. Place the back on the quilt top, right sides together and stitch a 1/2" seam around the 4 sides -- depending on the size of the quilt leave a 10" or larger opening since you will be pulling the quilt through the hole. Trim the seams by cutting away any uneven fabric or batting. Clip the corners by trimming off a little triangle. Do not cut through the stitching. Turn the quilt right-side-out, then press so the backing is not visible from the front. Fold and press the open edges inward a 1/2". Glue the opening edges together. Top stitch 1/4" from the edge to secure. If you want to secure the back to the quilt you can tack it or do a bit more quilting. I have also spray basted the batting to the top instead of quilting it, stitched and turned the quilt, finished the edges by stitching and either tied or quilted the quilt at that point.
I like this finish for quilts that are going to be used very hard.