The legal position of spaceflight and its associated emergencies is a very nebulous one in international law. The advent of manned spaceflight superseded a number of laws regarding the legality of overflight and the nature of sovereign territory. For example, the 1944 Chicago Convention helped establish the groundwork for International Civil Aviation Organization- ICAO which clarified a number of the basic rules for emergency landings and what constituted an infringement of national airspace. The ICAO established that the airspace above national boundaries is national territory, there can be no neutral crossing of this territory. Space vehicles sidestepped this legal precept. De Gaulle demonstrated this facet of outer space to Khrushchev when the latter went on a harangue about the Gary Powers U-2 incident at the Paris Peace summit. He told the Soviet premier "yesterday that satellite you launched just before you left Moscow to impress us overflew the sky of France eighteen times without my permission. How do I know that you dd not have cameras aboard while taking pictures of my country?" upon which Khrushchev flubbed his answer. As laws regarding space have evolved, a general consensus has emerged that space is neutral territory, unlike that of the atmosphere.

This of course poses the question about what about landings which the OP alluded to. As space vehicles transitioned between atmosphere and space, they occupied an indeterminate zone. The UN sought to clarify matters in the late 1960s with two international treaties that outline both the responsibilities of the state that launched vehicle and the wider international community as a whole. The 1967 Treaty on Principles Governing the Activities of States in the Exploration and Use of Outer Space, including the Moon and Other Celestial Bodies maintained the idea that astronauts and space exploration as a whole was a human endeavor that transcended national boundaries. Later, and more pertinent to the OP's question, the Article 4 of the UN Resolution for Agreement on the Rescue of Astronauts, the Return of Astronauts and the Return of Objects Launched into Outer Space established that in case of emergency, all signatories of the resolution would:

If, owing to accident, distress, emergency or unintended landing, the personnel of a spacecraft land in territory under the jurisdiction of a Contracting Party or have been found on the high seas or in any other place not under the jurisdiction of any State, they shall be safely and promptly returned to representatives of the launching authority.

Of course, these UN resolutions are dependent upon self-enforcement and whether or not either superpower would have upheld them is venturing into what-if territory. However, it is likely that they would have obeyed the strictures of these late 1960s agreements as they fit within the established paradigms of international rescue pioneered by the ICAO.