You really want country of origin to be other than UK. If you cannot go to Sweden to take advantage of the lower fare, at least go to AMS and start from there on the RTW. Then you can return to LHR and do flights like LHR-DXB later. You cannot re-enter the country of origin, so once you land in the UK, you cannot leave UK if they RTW ticket is bought there.

From ORD you can fly AA ORD-DUB-LHR for about half the price of ORD-LHR; or you can go to JFK and fly out of there (often a lot cheaper than ORD-LHR).