I'm speaking from ignorance, since I haven't lived in NY for 50 years . . . but I would be VERY surprised to see a response from Nassau County into Westchester.
Back in the fall of last year, Nassau and Suffolk sent swift water assets up to Westchester during Tropical Storm Ida. They staged at the Westchester Fire Training Center in Valhalla. I'm not sure if they were deployed into the field.
On the flip side, Westchester sent fire suppression and hazmat assets to Long Island after Hurricane Sandy. Units staffed firehouses as well as the Yonkers FD leading a hazmat task force collecting propane tanks that had washed ashore from Fire Island as well as mitigating heating oil spills from flooded heating oil tanks.
Both responses were triggered by activating the Statewide Mutual Aid Plan. A few years back, the Yonkers FD and several additional career departments were sent to Buffalo as a technical rescue task force when Buffalo were hit with over 6 feet of snow. Those units conducted structural assessments of unstable buildings that were damaged by the heavy snow load.
As far as Mount Vernon goes, it's not uncommon to see them operate with the FDNY on border calls. Usually, the FDMV tour commander and the FDNY Battalion will confer as to what resources are needed. However, their mutual aid plan only includes Westchester units for the most part. Requesting FDNY companies is not so simple because it typically has to be approved by either the borough commander or the citywide tour commander.