In addition to bundling up when possible, there were three things that would keep a sailor warm enough not to die of hypothermia, though certainly not comfortable.

First, men on a sailing ship move around a great deal. They are hauling rope, climbing into the masts, rolling sails, and otherwise constantly running around. The constant motion keeps the body warm.

Second, sailors often coated their shipboard clothing with a layer of tar. The tar would make the clothing somewhat waterproof and would keep out the worst of the damp.

Last, sailors generally only stood short duty watches. In English and American vessels generally 4 hours. Afterwards they would go below to warm up, get some food, or grab some blankets and sleep. Some captains would have the cook keep hot food available in the galley if conditions were safe enough.