It took two reads, but I think I get it. The judgement is immensely helpful.
Contrary to what you say (and assuming A is your client and B is the neighbour) it is very clear now to me (thanks to the court filling in the very blanks that we have been debating) that your client's neighbour can remove the roots as long as he doesn't rely on the law of nuisance or the TPO law to get access to your client's land to do so. That is the first point. Your neighbour can cut the roots from his side, with impunity, as long as they are causing damage and therfore constitute a nuisance.
The next step to consider is whether your client can rely onm the TPO laws to cut the roots from his own side with impunity to abate the nuisance to the neighbour. Again the judgement seems very clear, he cannot.
All of this is what we have suspected all along. Accepting the neighbour's clear rights and the client's clear restrictions, the hypothetical questions must be answered (i) is it reasonable for the Council to refuse an application to remove a tree that will be dangerous (and you would have to be satisfied in that regard) after the neighbour has removed the roots and support from his side and (ii) was it reasonable for the Council to refuse the application for the reasons it did?
I have before given my tupppenceworth that the refusal might be technically correct but churlish. I still think that might be the case and partly for me answers the first question (which will be the essence of the appeal debate if your client appeals). I think the answer to the second question has to be no, the Council has no locus at law to interfere with the neighbour's absolute right to abate the nuisance, and the TPO laws do not give the Council any additional locus to interfere or prosecute or to say how he should accommodate the nuisance.
So what can be done. The neighbour can remove the roots. He cannot be stopped or prosecuted. The tree will (if you are satisfied in this respect) be dangerous. It can then either be removed by your client with statutory immunity from prosecution or be the subject of a second fresh TPO application. If the second one is refused, the Council is responsible for compensation for damage caused by the falling tree.
The dilemma is brought into clear focus before a second application is made. If you are satisfieds that the tree, after root removal, is dangerous you should tell your client and recommend he exercises his right to remove it without further consent. Conversely if you are not satisfied, then you should apply and give the Council the option to refuse and to take on compensation responsibility (but only to the extent that damage was foreseeable and couldn't be mitigated by your client in other ways) or to approve removal if it concludes that the risk of damage and harm outweighs the public amenity value.
Is it not this very matter that has been anticipated and been the reason for the first application? Your client might be frustrated ant a refusal that ignores the inevitable issue, particularly when the Council has misguidedly refused on spurious grounds.
You have probably got no obligation to advise the neighbour, but your client would be at liberty to have you explain to the neighbour his position (without prejudice, of course) as regards abatement and immunity from prosecution. If the neighbour then cuts the roots, the tree must go if it's collapse is inevitable and imminent, and should probably be the subject of a second application if it is not imminent.
My advice to the client about whether to appeal now would probably depend on the neighbour's immediate intentions as regards root removal. If he plans to cut, the course ahead is reasonably clear. If he does not and plans to tolerate the roots, the problem disappears. If he wants to force your client to abate the nuisance by other means he, well, can't. If he invites you or your client to cut the roots at his yoyur client's expense but on the neighbour's land, the way ahead is just as clear as if the neighbour had wielded the saw, it matters not whose hand cuts, only who instructed it (i.e. the neighbour).
I hope my opinion and analysis helps. I long to hear the outcome. If your head was nipping before, I only hope I haven't made it worse.