I hold a PhD in political economy, and am outside academia; what route might I take towards researching for a PhD in canine psychology and behaviour?
Given that you already have a Ph.D., can you network with anyone to meet a mentor in this subfield? Their personal advice could help you figure out which formalities you can skip and what skills are essential. Do they see your work as providing transferable skills to their field? Would they recommend that you need more background in lab work, psychological theory, or something else?

While it's not normally a good idea for people to try to contact unknown professors merely to ask for help preparing for a later application, you've been socialized into the norms of academia (especially within the Russell Group) and may be able to do so in a targeted and effective manner.