In a hypothetical situation where ALL man-made structures on earth are destroyed (including servers, cell towers, radio antennas, etc.) but anything existing underground or in space (satellites) would remain intact, would any long-range communications (like from one side of the US to the other), like the internet, continue to function? For how long? This is assuming survivors bring cell phones, computers, radios, or other communication devices with them into underground bunkers, then emerge once the apocalyptic event has passed. Would they have any means of communicating with people thousands of miles away, given the proper equipment? If not, what is the furthest they could theoretically communicate?
Radio waves can be bounced off the ionosphere. Transatlantic communications were achieved as well as communication between North America and Hawaii with technology that is almost 100 years old.
Something would still exist, and could be expected to continue to exist for decades. Indefinitely if they could scrounge or make spare parts.
Ham radio operators have made portable setups to allow laptops to communicate together using techniques with global range. These schemes currently use some fixed infrastructure, but could easily be changed to be totally ad hoc.
Of course the number of people able to do this is fairly small (2-3 million hams and probably 10k's of digital users), and if all the servers are down there isn't much of an internet, but the old guys could still be trading emails world wide.
A slight change in the question changes the answer a little:
Short wave radios are fairly easy to make and have global range, most electrical engineers or amateur radio enthusiast should be able to cobble something together if they have access to any kind of electronic scrap. If they can bring a few dozen pounds with them into the bunkers it should work from day 1.
Can we have long-range communication? Absolutely! Short-wave radio will always be available, and communication satellites will be working for some time (at least months).
Can we have internet? Not really. While internet was conceived with a fault-tolerant architecture, it now relies too much on a certain set of backbone servers and links. If one of these servers or links is completely down, we can see large internet blackouts. If all of them go down (and stay down), internet as we know it would not be able to recover. In your scenario, if we have a surviving team of communications engineers who can start new servers and reprogram satellites, then there is a chance, but otherwise no, the internet is done for.
Iridium satellite phones should keep working for as long as the satellites remain on station without communication from ground control. The Wikipedia article says “Station-to-station calls from one satellite phone to another can be routed directly through space without going through an earth station.”
The first 'spark-gap radio' that started everything was really simple to make, and very low-tech. It used Morse code, and sent a signal across the Atlantic.
See this, for example.
So enterprising makers could cobble together some form of system, provided they have access to the information.
A long enough receiving antenna, a high voltage source, a few components.
Of course, this assumes there is someone listening as well as someone sending.
All they have to do is to go on the Internet for the instructions.
(Literally) Underground networks do exist today.
There are a number of fiber optic, undersea, and other form of underground communication links which exist even today. The original backbone of the DARPAnet (what is now known as the internet) had underground links as part of its' structure.
There is also the G.W.E.N system, which is based off of Nikolai Tesla's work. It is interesting to note that while the G.W.E.N system as was constructed by the U.S. operated using a ground-hugging wave propagation scheme specifically designed to function post-nuclear and/or large-scale EMP catastrophe, the original claim by Tesla was that his version could function by propagating a wave literally underground at great distances. The G.W.E.N network was never finished due to funding, and some of the structures have been possibly been re-purposed as cell phone towers.
If Tesla's claims were accurate, the towers could have been literally built underground, thus surviving a surface-level event.
Anything cell-based will be schtonk - cellphone systems rely on having more or less a clear line of sight (through air or light obstacles) to the next base station, that is why these are put on towers - and the base stations rely on a backhaul to function. This is either a cable link that will eventually rely on something above ground to function, or a directional, narrow beam radio link that relies on a partner antenna in a very defined place.
Satellite phones (not satellite internet links, too much aboveground infrastructure involved) are likely to work for a while - as long as the satellite does not need any attention from a ground control station to continue functioning.
As mentioned, there is always shortwave and longwave radio - however the bandwidth of communications possible this way is, just as it is with satellite phones, orders of magnitude too small to support casual, universal communications as you would get from the landline network, or the internet, or the cellular phone system.
The reason Cellphone systems, TV networks are done on VHF and higher frequencies (simply speaking, three and four digit megahertz) is exactly that such links can be made relatively directional and of limited reach, and that there is a lot of bandwith available. Literal bandwidth (range of frequencies you can use for your channel, eg 2401-2433MHz for 802.11b WiFi channel 1) and data bandwidth (how many bits of data you can get through per second) are interrelated. This is not limited to 1bit/s per Hz even if it looks that way superficially - but going above that limit means a lot of technical complexity (eg 90s 56Kbit/s modem links or DSL over phone lines having much lower bandwith - the modems are actually complex as all heck internally). The point is, establishing a 20MHz wide channel on shortwave would make the whole shortwave band unusable for anything else in the surrounding area, and would interfere with other uses in a far wider perimeter.
Direct satellite links have other gotchas: They need sophisticated equipment. They are expensive to build, so the actual comms bandwidth available to a whole ecosystem is rather small. And they are slow, not from a bandwidth but from a latency perspective - radio waves are still limited by the speed of light, and satellites are far away in the context of modern, rather interactive communications protocols.
Buried landline-type infrastructure could be made serviceable by whoever really needs it, as long as shallow underground structures (think manholes in the sidewalks) are unaffected, and the end points of the cables and any locations where active, underground equipment (repeaters etc.) belonging to them needs to be fed power are accessible.