Apply for the renewal of your wife's Dependant Visa straight away, and at the same time apply for one for your son. My expectation is that your wife's will be renewed within 2-3 weeks, and that they will issue one for your son at the same time assuming you can submit the necessary paperwork (essentially his birth certificate I guess).
Then when he has the Dependant Visa he can come to Hong Kong and you can apply for the British Passport here.
I do have to wonder why on earth you made this so complicated by having your wife give birth in Pakistan. If she had given birth here it would all have been so much simpler!