I imagine the quickest would be with QF/BA SYD-LHR-EDI. The QF schedule has a 1h45m stop in Singapore and a 1h40m transfer in LHR on the BA codeshare. Total time is 26h05m.
Going via USA I'm guessing the quickest would be with QF107 to JFK via LAX. Problem there is I think the only flights from USA to EDI are from EWR, requiring a nice quick drive through the quiet NY streets... SYD-JFK is already over 21hrs, so I think LHR or other European stops will be the quickest.
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