It could be either voice via interphone or hotline to ATC, or via a data connection right to the flight planning/tracking system used by New York, Miami and San Juan ATC. My point being, in the ATC world what we call ATM or Air Traffic Management the push is on to reduce voice communication and hence the opportunity for errors in communication both via radio and internally, CPDLC being an 'external' example to those who monitor aviation frequencies. That data link alone eliminates the need for New York Radio to even relay requests and instructions as the CPDLC link is to ATC, not ARINC. In your example of a request for deviation, there are pre-defined messages for exactly that, the request and the approval to deviate, with no need for readbacks in either exchange. Internally it could go either way (voice or data) between NY Radio and ATC when a non-CPDLC flight makes that request. My bet would be voice for short-duration changes like WX deviations. For a re-route, I'd bet more on a data link to the flight planning system.
FWIW, the inter-agency (international) network connecting ground facilities is called AFTN, the aeronatucial fixed telecommunications network. It was established at least (guessing here) 45 years ago, probably over 50 really, as originally a 110 and 300 baud teletype network for flight plan, NOTAM and other data exchange. It still exists today and includes drops at ATC, FSS, ARINC, SITA, military and aircraft operations locations globally.
Bob