So I’ve put new oil(Synthetic) in my ST as it was low but the car has been sat for 2-3 years and I haven’t changed the filter. While the car is running white smoke comes out the back, not a lot unless I rev it. Is this down to the synthetic oil I put in or the fact I haven’t changed the filter so old oil is burning. Or something else?

The oil should be synthetic so ok there

With being stood for so long if the engine wasn't started the cylinders might have lost the oil film lubrication which will cause wear when started again so it could be related to that. Or it could be moisture burning off after all that time standing

2-3 years it's probably just condensation if it's thin-ish white smoke.