I think it would depend greatly on how the birth went. If it is a natural delivery with no complications or stitches then very gentle exercises could be started as soon as mom felt up to it. Stitches take longer to heal, and the 6-8 weeks is recommended. C-section births take longer for recovery, but the actual time depends on the mother.
I was told to start the day of delivery doing kegel exercises. From there it was recommended to do no more activity than required to care for the baby. At 4-6 weeks they said I could start a light exercise routine and work up to my regular activity level. I made great friends with the yoga ball during that time. It made doing ab exercises a lot easier.