As much as I dislike Heathrow, you are probably better off flying from there. At best your flying time to Brisbane is going to be 23hours, with at least one stop. Every stop you add will adds three or more hours to the overall journey. A number of airlines will fly from LHR to BNE with only one stop, whereas you will always have three, possibly four if you fly from LCY. You're better off spending the time in a car or taking the Heathrow Express from Paddington.
Flying via the US isn't a good option - there is limited competition on the US-Australia route, so service and price suffers as a result.