You don't have to be an amazing sewer to replace elastic on a nappy. In fact, for a nappy with encased elastic, you don't need to be able to sew more than a few stitches by hand!
For a fitted nappy with sewn in elastic, though, while you will need a sewing machine, it's really not hard at all. Here, Vicki from Bubblebubs demonstrates how to replace the sewn in elastic in a fitted nappy like the Bubblebubs Bamboo Delight or Bam Bam fitted nappies. This will work for the Baby Beehinds fitted nappies as well.