I believe the Mustang S550 (sixth generation) built 2014 on had an oil cooler. Earlier versions including the 3.7 V6 2011-14 Cyclone engine which the 3.7 Roadster used have neither oil cooler for engine or transmission. No EGR either thankfully.
I think this applies to all trads ex factory excepting the 3 litre Roadsters undocumented oil filter heater/cooler which can't have made much difference.
I suspect this won't be the case with current CX series though. Certainly our 2016 F54 Mini with B47 diesel is part of a massive recall to replace EGR cooler due to faulty batch. Current situation is they are building up stocks to service the recall.