All of this is according to Wikipedia.
The first commercial telegraph was installed in 1845. Noncommercial lines were in place earlier than that.
The first commercial automobile appears to have been sold in 1886.
Early skyscrapers were built around 1890, and were about 10 stories tall.
Finally, the lightbulb was commercialized in 1880.
In AoL, the horseless carriage is a new invention, and lightbulbs were invented in the 20 years that Wax was away from the city. This pretty solidly puts Scadrial in the late 1880s or early 1890s both in general technological level and in electrical technological level. However, it's skyscrapers (which were expected to pass 50 stories) both exceed this timeframe dramatically. That means that on an Earth timeline, the telegraph should have been invented about 40 years earlier. However, technology is progressing faster on Scadrial, due to Sazed's copperminds. And of course, not everything is developing at the same rate, but I would assume that all electrical developments are tied together to some degree.
Looking at this, I don't think I can bend the timeline enough to avoid the existence of the telegraph based solely on technological development.
However, this doesn't completely eliminate the possibility of the absence of the telegraph. Perhaps early attempts to create a telegraph set things on fire, or exploded for some reason (I'm not even going to pretend I could come up with a reason why a telegraph would explode), and this set back telegraph development for some time. With such a centralized civilization, I suspect that independent development of technology was a lot less common than it was on Earth, and therefore a couple accidents could set back technology more than they would have on Earth.