Obviously, it could be embellished, but you're basically looking at something like this.
LHR-SYD-AKL-SYD-JFK(-LHR-JFK)-DFW-YYC//YVR-JFK(-LHR-JFK)-UIO-MIA-LHR
The JFK-LHR-JFK trips would be on your own dime. You could sub any city for JFK, but I think the NYC-LON flights are fairly reliably cheap(ish). The trip would be a five-continent.