No View it does not say acceptable. It says "However the HSA has reservations about both methods. In the case of neck dislocation, it is difficult to consistently achieve an immediate loss of consciousness. Similarly, decapitated birds may continue to show brain activity for up to 30 seconds after the cut is applied. As such, there is potential for distress, pain and suffering. The HSA recommends that neck dislocation should only be used in emergencies or for very small numbers of birds where no better method is available."
I would suggest that as all vets in the UK will perform euthanasia and it is therefore 'available' and as its recognised that cervical dislocation may to cause pain and suffering that cervical dislocation should not be assumed to be humane as a PP suggested or used as a first-line approach, but only in extremis. I'm afraid I can't speak for Hugh-Fearnley Whittingstall but I'd tend to place the advice of the HSA above his.
BTW I have a colleague involved in battery chicken de-populations, and yes they take them to the vet if euthanasia is indicated. But if slaughtering birds on a regular basis, an alternative would be to attend a HSA humane slaughter training day to learn correct techniques or to purchase a small stunner for domestic slaughter. Again both of these options are suggested on the link I posted and preferred to untrained neck wringing.
There are plenty of exotics vets out there though you may have to travel a bit further, just as you might to find an equine vet if you had a pony. But this is your responsibility as a pet owner. Owning exotic pets isn't a right, and it's important to ensure you can manage them properly before you do. There are a list of exotics vets available here www.bvzs.org/images/uploads/BVZS_Specialist_list_2015.pdf
In terms of the emotional component of pain - no one knows how pain is 'felt' even within the human species we have differing experiences of pain between individuals. But if animals are displaying behavioural responses to pain, physiological responses to pain and selecting analgesic drugs when painful conditions are inflicted upon them then I'm not sure what the basis is for suggesting that they don't feel pain? We do know that all vertebrate animals and likely some crustaceans and Cephalopoda are sentient and have the capacity to feel emotional states including pain so I'm not sure what you mean by higher centres being more developed? Or what you are basing this on? Emotions originate from the primitive part brain not higher centres as emotions such as fear and pain are essential in survival, and so whilst we cannot be certain that the experience is identical for every animal (in much the same way that I can't know that you feel pain in the same way as me), I'm still not seeing any justification for the withholding of pain relief for a severely injured bird.