As others have mentioned, you'll need a DONE4 for this trip. The lowest prices in Europe are still to be found in Cairo, although there's a whole clutch of countries that comes close: Turkey, Yugoslavia, Croatia, Poland, Romania and Bulgaria are about 50 GBP more expensive than Egypt at current exchange rates for a DONE4. There's not much experience yet with starting a RTW from any of these locations, but they would have the advantage of keeping a LHR-DXB-LHR mileage run open, which CAI precludes.

The routing for NRT, SIN, BKK and KUL is going to be tricky. The real problem is KUL, as I think you can only get there from HKG. You could do something like WAW-LHR-NRT-HKG-SIN-BKK-HKG-LAX-GIG-LHR-DXB-LHR-WAW, but I don't see how you could fit in a trip to KUL. Your best bet would be to buy a separate ticket for BKK-KUL-BKK, which may be cheaper than doing the SIN shuttle. You've still got plenty of segments to spare with the above, especially in the Americas. You could fly back GIG-SCL-LIM-MEX-LHR for instance and still have segments to spare. Or LIM-JFK-LAX-ORD-PHX-LHR to add some mileage. Unless I've miscounted, that would still leave three segments to be added... perhaps add a stopover in GRU? GIG is so much nicer, but if you're there anyway...

So then we have, assuming you start in WAW:

WAW-LHR-NRT
NRT-HKG-SIN-BKK-HKG-LAX-GIG (using the transit clause)
GIG-SCL-LIM-JFK
JFK-ORD-PHX-LHR
LHR-DXB-LHR-WAW