Hi Paddy,
You are correct, it is an non-interference engine (have snapped a timing belt myself in the past). Without seeing the car its hard to know what the issue might be but I'd get him to check the timing marks are lining up correctly, the camshaft pulleys are lined up with the marks and there are 19 teeth on the belt visible between the tops of the camshaft pulleys. If they are correct just start trouble shooting the usuals (the car has spark, fuel and the airflow meter is connected and working).
Heres a v.detailed how-to worth printing off for your mechanic :
http://www.miata.net/garage/timingbelt.htmlLet us know how you get on.