Assume for a second, that you have a pond. And a new type of algae has been introduced into the pond. Algae grows quickly, so let's assume a doubling time of a day. 24 hours. The concern is that this new algae is gross and smells bad and nobody wants to have a pond full of this disgusting algae. Unfortunately, treating the algae is expensive and nobody wants to treat the entire pond.
The question is: One week before the pond is entirely covered in algae, would enough have appeared that you would even notice? At a "gut instinct" level, we'd guess that perhaps a quarter or a third or at least a tenth of the pond would be covered in algae, but that gut level instinct would be completely wrong. Just 1.56% of the pond would be covered - right about the point where it becomes noticeable at all.
The point is this: information processing capabilities, globally, aren't just growing exponentially: the rate of growth is itself also growing exponentially. Just about exactly at the time where we notice actual, verifiable intelligence of any kind is just about exactly the time where we have to assume it's ubiquity.
Well, yes, but no. There's a major difference. The algae can self-replicate in every form required to spread. A computer AI cannot - it's limited to the hardware it was built on, and it would not be able to build new hardware to add to itself. Further, heterogenous processor environments (networks, switches, etc) would also limit it as it would have to be able to reach out across them and assimilate them into itself.
So no; it will be very unlikely that by the time an true AI of any sort at the lowest level of intelligence is created that it would be necessary to assume ubiquity or that a Skynet-like singularity moment will happen. To start, the AI would have to be far smarter than that to be able to move itself to another system. It won't just "magically" happen to be able to run itself on all types of computer infrastructure at the same time.
Previous discussions talk about the number of cross connects and how far away we are from the mark without commenting that the Internet itself allows for an infinite number of cross connects - my laptop can connect directly to billions of resources immediately with an average 10-25ms delay. Now, it's very likely that what is meant by "cross connects" in the context of AI is substantially different than the "cross connect" capability that global networking enables, but it's equally true that people generally fail at understanding exponential growth. It's why 401ks are so universally underutilized, why credit cards are such big business, and why the concept of the "singularity" seems like such hocus pocus at the gut level.
So there's another flaw in your logic. Yes, your laptop may seem to be able to connect to an "infinite" number of inter-connects, but reality is that it can't. For instance, each network connection requires a small chunk of memory to manage it, plus some more for the routing rules for the networking so it can get off the computer and out onto the larger network which then has to have the correct routing rules for it to be able to get to the Internet, assuming that the network is even connected in a way that it could reach the Internet (not every network is Internet connected). As a result Operating Systems impose a limit on the number of network connections; usually done through limit Open File Descriptors. For instance, Linux by default only allows about 4-5 thousand open file descriptors at a time for the entire system, not per application, due to the OS-level memory consumptions required to manage each - this includes all network connections.
Now, you'd say that the AI would "magically" overcome this by developing a new way to track it all, or just use a few at a time. However, that again limits it as it generally takes a lot more than 10-25 ms to establish network connections. Using TCP is a good example as it is the ultimate optimization of having a UDP-based protocol for tracking a connection state - that is, the typical work around is to use your own UDP protocol, but that only results in moving more memory into the application, not really overcoming the issue.
This again also does not overcome the issue that the AI would still be limited to the hardware available to it for learning - that is, the RAM and Processor and all the hardware in it; it would also be further limited to the environment (both OS and hardware) and knowledge it has about any given portion of hardware. It won't be able to magically rebuild itself for an ARMv8 processor when it was built on an ARMv7 as there are major differences between the two; or to go from a Power processor to an x86 processor. It won't be able to optimize itself to run at peak on the hardware without having been given knowledge of how a processor works.
All-in-all there are many many limitations that will prevent the spread of AI such that it will be well known that a true AI was created in some form long before the "singularity" moment occurs. And, if you really pay attention to most of the Sci-Fi stuff, the Sci-Fi stuff generally acknowledges that since it is the norm that a government or corporation or individual spent a long time developing the AIs, communicating with them, teaching them, in private before the "singularity" moment occurs - which is typically related to some preservation realization when someone wants to shut it down.