You can call AA's RTW desk and set it all up so that when you get to the UK, just go to the LHR ticket booth. Do NOT get it issued in the USA because you will pay the higher US price.
The official OW planner is a bit buggy, but you are OK with 4 segments in Asia.
It appears that you have 8 or 9 flight segments to use out of your 20 permitted. You can use up to 6 within North America and 2 or 3 in Europe. You could fly to Mexico or Central America, then instead of returning to the US, fly Iberia nonstop to Madrid and use up to 3 segments in Europe, but only 1 stopover. Or fly to any other non-UK destination from AA's USA gateways, eg ORD-FCO-MAD-HEL-LHR. And you don't have to return to EDI since surface segments within the country of origin are permitted. The Scottish may think that England is another country, but for this fare LHR and EDI are in the same country. Saves a segment for another use.