Presumably it would require legislative change to either create an appellate tribunal or direct appeals to the existing Upper Tribunal.
Correct, even if all chief adjudicators agreed with Hippocrates, ultimately only Parliament could bring about such a change. The obvious route would be an appeal to the existing Upper Tribunal, similarly to what's been done in Scotland.
As always the question will be cost, as appeals to the UT would either have to carry very significant fees to cover the costs of paying judges, admin staff, IT systems and so on, or some charging mechanism would need to be introduced whereby enforcement authorities get invoiced by HMCTS for the cost of such appeals. Councils won't want to pay anything, so the government of the day would need to impose a charging mechanism on them.