I've seen pros take a seat cover apart in about 5 minutes using a single edged razor blade. Something in the back of my brain tells me that if I try something like that at this particular stage of the game, I'm going to make sushi out of my fingers. So, keeping that in mind, I did it the old fashioned way, with an old-fashioned seam ripper. Basically, you just tuck the longer point of seam ripper into the seam, insuring the thread of the seam is tucked into the U-shaped channel of the seam ripper, and push. There's a bit more to it than that, like not cutting into the vinyl, but that's basically it. I took the seat covers apart in about 10 - 15 minutes apiece, including the time it took to stop and take pictures along the way.
Remember that I'm going to use this old cover as a pattern. SO DON'T GIVE IN TO THE TEMPTATION TO GET A SEAM STARTED, THEN JUST RIP IT APART! At the least you'll stretch the vinyl in a direction that you don't want to stretch it, and the cover you make from these pattern pieces won't fit. At the worst, you'll rip big chunks out of the cloth or vinyl, and have to guess when you go to trace the damaged piece for your new cover. This is maybe 20 minutes out of your life - take your time and do it right.
Working slowly and methodically around the covers, I got them apart with no trouble at all - other than getting a bit greasy cutting one corner of both the bottom and the back. More on this later.
Picture 1: If you've never seen one before, this is a seam ripper. Your wife/girlfriend/mother probably has one. If not, they're about 75 cents at Wally World. You can use a single edged razor blade if you'd like. I'm going to bounce back and forth between the two until I get better at controlling the razor blade without cutting up either the seat or myself.
Picture 2: This picture shows how the seam ripper goes into the seam. The longer pointed tip is tucked into the seam, and the thread sits in the u-shaped channel. A firm push will cut the thread until it hits an obstruction, usually the vinyl. You can cut the vinyl with this, but not as badly as you can with a razor blade.
Picture 3: Tucking the seam ripper into the seam between the insert and the weld and taking my time, I had the back apart in about 10 minutes. The bottom took a little longer than that because there are more seams to cut. Once the seam between the insert and the welt was cut, the welt pulled right off of both pieces.