the passport control is the worst bottleneck section. i'd say it's even worse than the traffic on the streets! i know you mentioned you're flying economy, but just to give you an idea what happened to me... i flew from IST to DXB on paid C, last week on a 7pm flight, arriving around 1am. if you fly on first or business, they give you a fast track "ticket" so you can avoid the regular lines at passport control. you'd think arriving 1am in the morning and with fast track you'd be able to wiz through... YEAH RIGHT. lines for all sections were super long and super slow.

your best bet is the e-card, issued to residents. long story short, stay in the airport. look around their duty free, although i think it's not that good compared to LHR.