Well, here's one option. I'm not sure if it's the best (specifically the inbound, but it's one of few itins that didn't require an overnight somewhere).
This requires booking two one way awards, otherwise they'll charge you for a RTW award.
3/22 NZ1 LHR-LAX in C
3/22 NZ1 LAX-AKL in C
4/22 TG492 AKL-BKK in C
4/22 TK61 BKK-IST in C
4/23 TK1759 IST-LIS in C
4/23 TP356 LIS-LHR in C
If you don't mind one overnight in BKK, you can do AKL-BKK-LHR on TG.
Chris