Air carrier Captain reported experiencing fumes in the cockpit on departure; resulting in the flight diverting to the nearest airport where they landed safely.

Narrative

After lengthy taxi and deicing; fumes in cockpit immediately after rotation. Caustic smell was burning our nasal passages and we donned 0xygen masks for entirety of flight. Lost ability to transmit or receive radio comms on hand off to Center due to precipitation static. Received and accepted CPDLC message to climb to FL360. At roughly FL200 we were able to regain comms with ATC and requested decent to 14;000 to troubleshoot the issue and burn more fuel for MLW (Maximum Landing Weight). Unable to contact dispatch via Sat phone after multiple attempts. Requested weather at nearest suitable airports from ATC and requested priory handling . ACARS messaged dispatch that we had fumes in the cockpit; requested priority handling and proceeding to ZZZ. Completed Smoke/Fumes checklist and Overweight Landing checklist. Configured early with flaps and gear to burn more fuel and landed under MLW. Tower advised they hadn't yet plowed the taxiways and to expect to turn off at the end of the runway. There was apparently a controller handoff and on rollout the new controller requested that we exit at the next available taxiway and hold short of taxiway 1. We made a right turn on the next available taxiway; I believe it was 2. The controller then asked why we had turned on that taxiway; saying that the taxiways were closed because they hadn't been plowed yet. We taxied behind a follow me truck to the ramp and after communicating with ARFF (Airport Rescue and Firefighting) shutdown engines.