Sorry to hear about your injury. Assuming that this is your lower lip, the injury would be to the inferior alveolar nerve, which runs inside the mandible. I am assuming that the damage is probably from the surgery rather than the anesthetic injection. Either way, it would be reasonable to seek an opinion from an oral surgeon with experience in nerve repair if you do not have improvement by the 8 week mark. You may want to start the process now, as it can take some time to find a surgeon with experience in this area, and to get an appointment.
I am not familiar with the literature specifically about nerve damage from apicoectomy. A quick literature search tonight did not yield anything. The inferior alveolar nerve injuries from tooth extraction seem to do pretty well with surgical intervention. Not everyone recovers, but a majority recover some sensation if the surgery is performed within 6 months (and most like to do it around 3 months).
Again, I am mostly extrapolating from the tooth extraction nerve damage data, and assuming that apicoectomy would be similar. If you find out otherwise from a knowledgeable oral surgeon, be sure to let me know.
If your plastic surgeon has significant experience in oral surgical repair and nerve repair, he may be as good. Many do nerve repair elsewhere but not in the mandible.
Did your first dentist imply that the injury was from the injection or from the procedure?