What's taking the time? I do mine by hand - find using the machine really fiddly.
This is how i do it: I stitch one end of the new elastic onto the end of the old elastic, then stitch the other end of the new elastic in place. Then when you unpick the other end you can just pull the new piece through, unpick the old bit and stitch it in.
Did you get them finished? Can I ask you what length of elastic you used in the legs? I have a load that need changing and although I am finding it quite straightforward I am not sure another the length as it seems to vary!