Fuel load for Birmingham-Geneva is in the region of 8 or 9 tons. The tanks on a standard A319 take 19 tons. Sounds like the plane was fuelled to full by mistake.
Refuelling today is done by the bowser drivers, not by engineers or pilots. They set the refuel panel, and then pump fuel until it stops. A power interrupt during refuelling can make the preselect figure to disappear and only an alert refueller will notice this.
At my airport, unloading 10 tons of fuel would mean finding an empty bowser. The bowsers are always kept full, so the fuel company would have to use one until it is empty. The bowsers hols 40000 litres, so unless you had a widebody waiting for fuel (not many in BHX) it would take 5 or 6 refuellings of an A320 to accomplish, before you could start the defuel. Takes awhile.