There is no similar protocol among non-NATO submarine-operating nations in the Atlantic. We do have similar protocols with certain nations in the Asia-Pacific region to avoid mutual interference when submarines are operating on exercises with each other, and that sort of thing.
With the Russians, the answer is no. They do operate an underwater telephone communications device on an international frequency, usually for emergency communications, but beyond that there is no process in place to allow a Russian submarine and a NATO submarine to operate in the same vicinity at the same time.