Infrared boost phase detectors on satellites have been around since the 60s. The US and Russian detection systems, at least, certainly know the origin of any ICBMs. There's not many parties that don't have that data and also have counter-measures.
Presumably you see where they take off and where they are headed for its pretty straight forward. Current ICBM's seem to have "relatively" minor down range maneuverability so once you're ballistic you can plot a big circle of where you can land. Is that near a spaceport? cool. Is it near a missile silo? better check into it.
Bombers are the same way, there is nothing that prevents building a nuclear bomber in the shell of a 747 and calling it a cargo plane on its transponder.
You just build second strike capabilities and call it a day. If {Russia, U.S.} gets off a first strike, then {U.S., Russia} can still get off a second strike, therefore neither will strike first.
Well, that's one option. Russia will need an early warning network to rival the American one either way, and given that and proper flight path registration... maybe there would be no confusion. Maybe.
Something like a rocket version of ADS-B that feeds to a central data collection point, from which nation-states that are nuclear powers (permanent UN security council members, for the most part) can receive a read-only data stream of rocket traffic.
Would require rockets with speed/altitude limit uncrippled GPS.