It will depend on your route...If you went from the UK by BA via Australia, it could take up to two-and-a-half days with a compulsory stop-over in Oz, due to timetables.
The quickest is probably 23 hours and 45 minutes from Heathrow to Nadi International Airport in Fiji, via, believe it or not, Tokyo! This is the Great Circle route over the North Pole and therefore the quickest. The time actually includes transfer time between planes in Tokyo.
The cheapest would be a round-th-world ticket, and would involve travel via the US to Fiji, then back via Australia or New Zealand.