I will mostly speak from the Lithuanian point of view, as that is the one I am familiar with due to my studies and expertise.

How did this unfold on a practical level?

In practice, the recognition of state continuity of the Baltic States by the US and numerous other (primarily Western) countries was expressed as such on a practical level:

• First, and perhaps most importantly, the US and other countries not recognizing the annexation of the Baltic States allowed the continuation of the diplomatic service of the Baltic states. Representatives of the former republics in the US, Vatican and elsewhere were allowed to retain their posts and offices even after the war, and were an important factor in organizing the Baltic exile community and campaigning for the interests of their occupied homelands. In Lithuania's case, this was organized by the Lithuanian Diplomatic Service, initially chaired by minister plenipotentiary to Italy, Stasys Lozoraitis, who was appointed as the chief of the LDS a few weeks before the occupation of Lithuania. In some ways, the LDS worked as a de facto government in exile of Lithuania during the occupation (although it was never recognized as such de jure, and was not universally recognized by Lithuanians in exile)

• Second, Western countries froze assets previously held by the governments of the Baltic states and refused to transfer them to the Soviet Union under the argument that the Baltic socialist republics are not legal successors of the interwar states. In Lithuania's case, the most notable assets were Lithuania's gold reserves, saved in banks in France, the United Kingdom and Sweden. Of the three countries, the UK and Sweden paid compensation for the previously stored assets to the reestablished Lithuanian state, while France managed to retain the Lithuanian gold reserves in their banks and returned them.

• Third, passports of the interwar Baltic republics and passports distributed by the exiled diplomatic service were recognized in Western countries. Refugees from the three Baltic states were also not repatriated to the Soviet Union from the displaced persons camps in Allied-controlled Germany in the immediate aftermath of World War II, again, under the argument that their state of origin remains the occupied Baltic republics, not the Soviet Union. Many of these refugees were intellectuals and former politicians who would have been repressed in the Soviet Union, and could instead campaign for the interests of their home country in exile.

Did the US bring this up in diplomatic meetings (e.g., "Hello, Ambassador." "Hello, General Secretary. When are you leaving the Baltics?")?

I'm not very familiar with the subject of diplomatic relations between the United States or Soviet Union, so I cannot offer much specific facts (especially as one would have to delve deep into the diplomatic communication to make any judgment), but when the fate of Eastern Europe was being discussed, especially at the very end of the Cold War, the fate of the Baltic States would become a topic of contention. The Malta Summit of 1989 would be a good example, as, to my understanding, George Bush actively pushed the topic of the Baltic states to Mikhail Gorbachev during the meeting (though my source on this is an interview with Vytautas Laurušas, who was, at the time, a representative of the Congress of People's Deputies of the USSR, and a contemporary of Gorbachev, so not exactly an academic source - judge it as you will).