Why would they want to do this? For LHR-bound passengers, it's a longer flight than via DXB, and a USA transit is a total pain in the cough. There's already lots of competition on LAX-LHR, so I doubt there's too much money to be made on that segment.
It works for NZ. Sure, AKL is a little closer to LAX compared to SYD/MEL but it could work for QF as well, but SYD/MEL-LAX is around the same distance as SYD/MEL-DXB.
The advantage is that LAX has much greater O&D traffic than DXB. It has more O&D traffic than any other airport in the world. QF also already has a hub of sorts at LAX with daily flights from MEL and BNE, and 2x daily from SYD. Frequencies could increase if both North American and European traffic were routed through LAX and conceivably there may be enough traffic to warrant a flight, say, PER-ADL-LAX.
On the other side, LHR is already served in OneWorld by BA and AA. UA, NZ and VS also service the route. There may be other opportunities available to QF apart from LHR if they can secure traffic rights. For instance, the 500,000+ passengers p.a. to CDG are only served nonstop by AirFrance and Air Tahiti Nui. The only carrier nonstop to FRA is LH.
I'd fare QF's chances of competing against AF, UA, BA and AA any day over competing against the likes of EK, CX, SQ, MH, EY, QR, TG, CZ, MU... Arguably QF would be the premium carrier there.
The disadvantage is that LAX-LHR is around 2000mi further than DXB-LHR. If the load factors were greater, the extra costs of fuel would be offset. Transit through the USA is also a pain in the behind, but NZ manages it just fine, and indeed it seems to work better for them overall despite the smoother transit their passengers had via HKG before they axed that route.
Wouldn't be surprised if QF were looking at this behind closed doors, as it looks like the EK partnership hasn't exactly delivered rivers of gold to QFi.