My guess, and it's just that, would be SYD-LHR - 21.5 hrs, LHR-SYD - 19.5-20 hrs , SYD-JFK 18hrs, JFK-SYD 19.5-20 hrs. Give or take. So in theory possible with 4 (that would allow 3 x 2 hr turn arounds and 1 x 11 hr turn around). But you've got have maintenance, allow for slot times and also operate at commercially appealing times, so 5 would be the absolute minimum I'd assume.
Times guesstimate based on SYD-DEL + DEL - LHR and SYD-DFW + DFW-JFK flying times. Block time is obviously greater than flying time, but there is also an extra ascent and descent when adding the two figures to make up for this.