The foam shouldn't have to be glued to the burlap, once you got your cover on and hog ringed it's won't go anywhere. And you shouldn't really need any cotton around the wire, the foam comes down around this, doesn't it?
When I reupholstered my -69 bench, I didn't do anything to the burlap or the foam, it was still in good shape. The burlap was already hog ringed in a few places though. I put a heavy (1/4") felt between the burlap and the foam (the "bun") and hog ringed it around the perimeter of the seat. Then I replaced the top cotton pad with 1/2" foam (spray glued to the old foam). Got the material from a local upholstery shop.
Don't know if the buckets are built the same way, but I hope you paid attention when you took them apart
I used PUI covers, and they look pretty good (there's still a pic somewhere in this forum). I found it easier to hog ring the back first, then turn it upside down and press the frame and the springs down with my knee while stretching it around the bottom of the frame. Again, don't know exactly how the buckets are made, I have seen seats with hooks punched out in the frame, but mine were hog ringed. If the covers don't have the channel for the wire, maybe it's just stretched around the front part of the frame? Note that the covers don't come with wires, you have to reuse the old ones. Look at them and see if there's a channel sewn into them, and compare to your old covers.