Is the thread fine or coarse, to the eye?
If the mounting is of recent manufacture, it is likely to be Metric.
Have you tried measuring the OD, and then checking the pitch with Thread Gauges?
Don't be surprised if the OD is just a little undersize.
I am discounting the threads being UNC or UNF ( 3/8 UNC would be 16 tpi and 3/8 UNF would be 24 tpi and 60 degree thread form instead of the 55 degree of Whit form threads. )
The results ought to give a fair indication of what the thread is. Since you mention 3/8 BSW, could it be M10?
If the thread happens to be 3/8 BSF the thread will look finer than M10, being 20 tpi (0.050" or 1.27 mm pitch ),
M10 will be 0.393" diameter as against 0.375" diameter, and 1.5 mm pitch against 16 tpi (which will look similar, at 1.585 mm but be different thread form )
If you have 3/8 BSW and M10 nuts available, trying to screw thew onto the male thread will also give guidance.
Howard