Honestly, most economy passengers making that trip would shower in Scotland and resign themselves to the fact that their next shower is going to be in New Zealand. I know it's a long trip, but that's what most travelers on this itin would do.
Yep. There used to be a cheap-ish shower option in Singapore but the rise of Emirates etc on the Australasian routes means you're often skipping it in favour of Dubai.
You've still got a 14hr flight to Sydney and a 5 hr flight to CHC ahead of you so the £50 Dubai shower is too early in the trip to make much odds anyway.