I wonder if a mechanical watch could communicate something via radio with some clever placement of magnets and copper on the movement via Faraday induction. Imagine movement that encodes a simple BT handshake. On the more science fiction side, a very tiny Difference Engine that fits on your wrist (I am reminded of a Young Ladies Primer from The Diamond Age, where the compute was nano-mechanical).

It does allow you to determine your longitude. So just run East or West, I guess?

Latitude is easier. Bring along a 5th-century astrolabe and you're all set.