Sorry to hear your tale of woe, Roz. They have clearly got confused with the extremely complicated Passport Agency system. If you (the mother) have a British Passport then the number and issue details need to be added to the form, and it doesn't need to go anywhere. If you have a non British passport, then it usually needs to be included as you child's nationality and eligibility (or not) for a British passport may be decided depending on what is stamped on yours (e.g Indefinite Leave to Remain etc.)
These are the things that will take far more training than any of those staff will have received, and no doubt test your patience!
Serious suggestion- ring the PO helpline to complain. They do follow these things up and it should prompt a refresher session for their staff.