If the parts finish or accuracy hasn't degraded, or the noise doesn't suddenly get worse, just complete the job, then deal with diagnostics.
If you have a gearbox, noise can come from there too. Run the machine at high "low range" RPM and see if a similar sound is present, then you have a good idea it's the 'box, not the spindle.
You can also use a mechanics stethoscope, or (careful!) something like a long screwdriver with the blade held to parts of the spindle, and the plastic handle held to your ear to try to pinpoint where the sound is coming from.
More advanced methods include using a accelerometer or acoustic pickup to find extra vibration and noise, but these are best when also used with a new or known-good spindle, so you can track changes over time.
For more information search "spindle bearing diagnostics" (no quotes) on Google.