They said they could see a tiny bit of exhaust smoke and then subsequently tried telling me because of the age of the car (it's a 2005) that it was most likely the valvestemseals.
As the oil smoke is coming out of the exhaust, that suggests to me that the oil is getting into the cylinders - the most likely causes for this are a blown headgasket, failed valvestemseals, or failed piston rings.