It's hard to get an exact figure, there are fuel prices, the time to taxi, gate charges, medical charges for the removal of the person on the plane, staff costs, and more. I can't guess at most, however from a fuel cost perspective I know for the 747-400 at LHR many airlines use a figure of 1000kg in planning for APU and taxi to takeoff. An A380 would be more, I'd ball park a figure of 1380kg for the airbus based on the MTOW difference (a rough approximation). 1380kg is about 1750 liters, at about .50 GBP per liter that's about 875GBP, or about 1150USD at today's conversions. So the fuel alone would cost somewhere around a grand, I'd imagine you could comfortably double that by the time all is said and done.
As for passing the cost on to the customer I've never heard of that being done for a genuine illness, or even a suspected illness. I don't know if they could if they wanted to, however I cannot imagine them trying it, the reputational fallout would be enormous.