Dear god, don't take public transport (Piccadilly line) with 3 suitcases! Heathrow Express may be more tolerable but Paddington is still a long way from the Docklands. Carrot Cars sound like a good alternative though I don't have any personal experience. I get daily spam from private hired car services so you shouldn't have any trouble finding one (on Google) at an agreeable price from Heathrow. Just make sure you know exactly where to meet the driver as the area outside the arrival door can be a confusing place.