If you would like to use your UA miles for this flight, you will most likely be flying on *A partner Turkish Airlines(TK) for a portion of the trip, however you may also fly on Lufthansa (LH) and EgyptAir (MS). Most, but not all routings go through Istanbul (IST) on TK. There are some sample routings below.
Outbound
LAX-LHR(UA) LHR-IST(TK) IST-BGW(TK)
LAX-LGW(UA) LGW-IST(TK) IST-BGW(TK)
LAX-IAH(UA) IAH-IST(TK) IST-BGW(TK)
LAX-MUC(LH) MUC-IST(TK) IST-BGW(TK)
Return
BGW-IST(TK) IST-LHR(TK) LHR-LAX(UA)
BGW-IST(TK) IST-IAH(TK) IAH-LAX(UA)
BGW-CAI(MS) CAI-LHR(MS) LHR-LAX(UA)
I was able to find Saver Awards in Economy for my sample search. With a Saver Award, it will cost you 42500 miles each way, or 85000 miles total. Have a look on the UA website and search for award travel. I strongly encourage you to plan your travel around the saver awards if possible, as a Standard Award ticket will cost you 85000 miles one way or 170000 miles total.
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You can also use your AA miles on this trip. I had trouble finding availability on the AA website, so it may take a bit more searching to plan your trip. To use AA miles, you may fly on AA, British Airways (BA) or Royal Jordanian (RJ). Some routings that I found:
Outbound
LAX-LHR(AA) LHR-AMM(BA) AMM-BGW(RJ)
LAX-MCO(AA) MCO-JFK(AA) JFK-AMM(RJ) AMM-BGW(RJ)
Return
BGW-AMM(RJ) AMM-LHR(BA) LHR-LAX(AA)
BGW-AMM(RJ) AMM-ORD(RJ) ORD-LAX(AA)
For the Saver Awards that I found, it will cost 45000 miles each way or 90000 round trip. I had difficulty pricing the ticket for a Standard Award.