How many sensory neurons are in the human body?
This is a really interesting question.  However, I'm going to try to address the context and update sections of your post rather then attempt to quantify all afferent signals in the nervous system.  This is because quantifying those signals would be an exceedingly difficult task.  It's not only logistically difficult to measure on an individual level, but to say anything definitive regarding the entire species would be superfluous.  The number of signals at any one time is constantly in flux, as not every sensory receptor generates an action potential at the same time.  Even in instances where similar or "competing" receptor types are simultaneously stimulated, those transmissions will be modulated and/or pathways suppressed.  A good example can be found in this editorial article[1] on pain, where inducing painful stimuli can alleviate an itch sensation.  I reference this to illustrate that sensory signals are not entering the brain unchecked. In the most simple of terms, they are heavily regulated.  

Based on the context on your question, it seems what you really want to know is what is the maximum threshold.  In an artificial nervous system, how many signals could we pump into it before it becomes inefficient or overwhelmed?  However, what makes the nervous system unique is that it's a living system that adapts to stimuli.  When there's more signals then it can handle or process, it regulates them.  You can read more about the process and its importance in this article on sensory gating[2].  For this reason, a healthy brain that isn't being artificially stimulated shouldn't become overwhelmed.  In fact, this process makes it far more efficient then any super computer.  A fun example of this was the six chess games between Kasporov and Deep Blue in 1996.  Think of the sheer number of moves Deep Blue could calculate compared to Kasporov: hundreds of millions of moves per second.  It lost that match.  Computational capability isn't conditional on the number of inputs the system can receive, it's how that information is processed, utilized, and applied once it's in the system.

I know I didn't answer your question in regards to a hard number, but hopefully it helped give a different perspective.  Please let me know if you'd like anything further clarified. 

[1]: Focus on pain.  Nature Neuroscience 17, 145 (2014). doi.org/10.1038/nn.3644

[2]: Sensory gating, inhibition control and gamma oscillations in the human somatosensory cortex.  Scientific Reports 6, Article number: 20437 (2016). doi.org/10.1038/srep20437