One of our members, @GrnyAny did a very detailed post about vagus nerve symptoms which were resolved with surgery, you might find it helpful:
Successful Surgery for severe Vagus Nerve Compression - General / Eagle Syndrome Stories - Living with Eagle
Most doctors do monitor nerves to make sure that they’re not being stressed during the surgery too much, but I don’t think that includes the vagus nerve. Hopefully in removing your styloid that will be enough to stop any compression or irritation of the vagus nerve. But be prepared that you may not feel instant relief, as nerves can take months to heal, and it could be if the nerve gets irritated during surgery, it could make those symptoms flare up & worsen temporarily.
I’ll pray for good results from your surgery, Dr Cognetti is an experienced doctor & has done many successful surgeries for members 
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