would be handy if he gave you asn emmissions printout at mot, let us know the figures when it passed.
beyond that, could be a number of reasons
a sluggish oxygen sensor, heater element within sensor isnt up to scratch, so has to wait for exhaust to be hot enough for sensor to work properly.
wrong grade oil in engine, not as bad as that sounds, for eg...if engine ideally requires 5w/30 oil, but been filled with 10w/40 oil, wont cause any harm, but will effect emmissions
coolant temp sensor is sluggish...fools ecu into thinking engine is cooler than it actually is, so increases fuel untill temp is up.
or might even be it just needs an oil change
those are most common reasons, none of them will give a fault code unless they fail completely....
if you know anyone good with diagnostics, reading live data will show up what is happening