I managed to get the following itinerary to price up for the dates you want using the on-line booking tool on oneworld.com
TYO-xKUL-CGK-xSYD-BNE-xLAX-PHL-LON-xAMM-JED-xAMM-LON-SIN-TYO

Notes:
1. TYO-JKT via KUL flying MH. The tool failed to price the itinerary if I chose the JL non-stop.
2. No direct flight JKT-BNE, so via SYD
3. No direct flight BNE-PHL, so via LAX
4. No 'D' availability on the BA non-stop LHR-JED, so RJ via AMM
5. Same situation for the BA non-stop JED-LHR, so again RJ via AMM

All-in price: JPY 1,280,390
Comprising:
Base Fare: JPY 936,500
Add-ons: JPY 343,890
of which surcharges total 164,420

ETA
The base fare will be the same no matter which airline you choose to ticket with, as will the various government taxes and charges
What might differ is the airline surcharges

You could ticket the above itinerary right now using the tool. I took the itinerary right through to the payment page. There was no indication of which airline would do the ticketing, so I presume it would be done by Malaysian (the first carrier on the Interary)
[often such itineraries are ticketed by Qanta, but this would be clear on the payment page - price converted to AUD and the Qantas logo clearly displayed.)
You can book with the AA around-the-world desk (tel: +1-800-247-3247} provided either or both of the flights BNE-LAX and PHL-LHR have AA flight numbers
You can book with JAL, if the first flight is JL