Considering long wave radio transmission be pretty much 1910s, see Popov's experiments, and the fact that long wave radio signal can wrap Earth, it's pretty possible that if one side rediscovers radio, the other side would be technically able to receive those transmissions. Also the ability to transfer orders fast is what actually makes empires stand and not crumple under their own weight, thus reinventing radio communication is desired by all states, perhaps they could even coordinate research for the benefit of all sides, as this tech is pretty fundamental. The same applies to the leftover community on the other continent, so if both sides manage to initiate technological revolution, they are bound to eventually come into radio contact with each other.
The possibility is there that they have too differing languages to not being able to understand what does the other side transmit, yet locating the source of a strong radio signal is possible with a little advance past initial discovery (modified radars), thus in a matte4r of decades either side would find out that the source of transmissions they receive is off their charts, so an expedition would ensue by either side, whoever thinks that contact with the outsiders would benefit them.
Also you pretty much forget that there was an epoch of Great Geographical Discoveries, that sufficiently predates inventing radio. In case the re-emerging states would desire expansion, they will commence a task to their fleet together with local shore control, in order to find any lost artifacts in the ravaged wasteland elsewhere, those that would embark on an expeditions to seas unknown might hit the forgotten continent as they collect maritime data on currents, winds, islands left etc.
To answer your title question
You asked "Why might there be no communication between continents for centuries after a civilization collapse?" - the answer is pretty simple, it took more than centuries for the remaining population to reach medieval tech level, as people were concerned with survival primarily. Having electric artifacts left over from pre-war times doesn't help unless you've regained knowledge of electricity, yet the supply chain on any modern power plant (even solar, even water-based) would not survive the nuclear war, thus there will be only remnants of stuff that was using electricity, without means to power them. (Some machines would be reworked into being powered by horse etc, yet these deteriorate too, and most easily accessible resources would be excavated, leaving no easy access to copper/iron/coal/oil aka energy sources, thus eventually all artifacts would become dust)
Seeing a satellite launched? Likely won't be noticeable
There are quite a lot of satellites in orbit currently, and while those that are on LEO would descend by drag while the civs would recover from post-war apocalypse, those that are located on SSO, let alone on geosync, won't, and those on SSO are visible at night as moving star-like objects. So people would get accommodated to seeing some moving stars, perhaps would use them to measure time, provided they would differ one from another and there isn't too many of them, and seeing another moving star might well be missed, unless the satellite would go "beep beep" and trigger radio receive across the globe. Astronomers of the mainland might be able to detect a new satellite and initiate a global outrage of "HUMANITY IN SPACE ONCE AGAIN!", yet should the mainland still be in 1500s due to technilogical drop, they won't really bother.
Also a frame challenge
"Being spared from the war, the smaller continent recovered faster, so they were the first to launch a satellite after the apocalypse" - since the pre-war civ was at least at 2000 tech level, there have to be major levels of goods transfer, at least for the power that had that colony on the small continent. As colonies usually go, the high-level industry was moved off the colony, leaving them dependent on mainland's production, in order for them to not revolt and gain independence. There might be other methods of keeping the colony in line, but all involve inducing dependency on the mainland and transfer of goods and personnel both ways. Then the war boomed, the mainland gets bombed, the colony gets abandoned, and then nuclear winter struck, leaving people in the colony without tech support for whatever installments of high-tech (power plants, computer centers, energy grid) they had, making them too go black. And getting out of black would prove very hard, especially with reduced population and deteriorated agriculture and general nature crippling. Even spared by war actions, the colony might get extinct instead of remaining afloat and with electricity.