I am a math PhD, not a history PhD, so caveat lector.

There is a great book on Fourier Analysis by T.W. Körner. What makes it so special is that it doesn't do a modern presentation of the subject. It's fairly unique among math textbooks in following the historical development of the subject, including chapters that are about historical episodes which the subject affected.

Chapters 65 and 66 discuss the laying of the transatlantic cable(s). There were a few misteps. The first couple never made it all the way from Britain to America. In 1858 a cable was laid which spanned the Atlantic. It took 16.5 hours to transmit a 99-word message from Queen Victoria to President Buchanan in 1858. The attempts to increase the transmission speed soon caused the cable to fail entirely within weeks of its initial operations. But even in those few weeks during which this faulty and slow cable worked, brought some major changes to news reporting.

In chapter X of The Story of the Atlantic Cable by Sir Charles Bright (a lot of people who worked on this project were very quickly knighted, and it is a major reason we know William Thomson as Lord Kelvin) recounts early messages to the AP about news in Europe, Egypt, and China, and how global news was now published in American papers the same day they occurred.

There was an immediate financial benefit to the British empire as well. A military unit in Canada was set to travel to India, but the situation no longer required their transport. The counter-orders would have arrived too late if delivered by traditional mail. By cancelling their mobilization via telegraph, British authorities estimate they saved 50,000-60,000 pounds. At that rate, the 1958 cable would have paid for itself in seven months. And this was the inferior cable which failed almost immediately. I am of course not pleased that the transatlantic cable enabled empires to more efficiently and effectively colonize and oppress people, but it is significant.

There was finally a long-lasting successful attempt in 1866, which could now do 8 words a minute; almost two orders of magnitude faster than the 1858 cable. In 2018 there was an American Economic Review study (which is publicly available!) which estimates a long-term gain equivalent to 8% of export values. I would hope someone here with greater ability would be able to shed light on how substantial this is. But it does seem the study is only on the benefits of reducing information disparity as applied to trade. And not all potential benefits economic benefits.

I'm not sure if this is the kind of social impact you have in mind, but the Victorians had a cultural fascination with telegraphy! I wrote quite a bit about this for my masters thesis in media studies.

Basically, the spread of telegraphic communications technology helped give rise to the spiritualist craze. People were really excited about the potential of immediate long-distance communication, and of what other powerful feats could be accomplished through electricity. Morse’s telegraph seemingly proved that science and technology could make information travel independent of bodies, transcending boundaries of time and space; and if it could do that, why wouldn’t the boundary between living and dead be next?

The spiritualists believed that electric currents might open up similar possibilities for incorporeal psychic transmissions. Tom Gunning writes, “Spiritualists embraced recent scientific devices, such as telegraphy and photography, both as tools for conveying or demonstrating their ideas and as central metaphors for their communication with the spirit world.”

This association was so strong that the methods and devices used to contact the dead were often referred to as “spiritual telegraphy,” which was not merely metaphorical language, but a genuine reflection of the spiritualists’ belief that what they were doing was of the same discipline as inventors like Morse and Edison. Jeffrey Sconce writes: “more than a metaphor, the spiritual telegraph was for many an actual technology of the afterlife, one invented by scientific geniuses in the world of the dead for the explicit purpose of instructing the land of the living in the principles of utopian reform.”

Even Thomas Edison, towards the end of his life, was interested in telecommunications technology as a means of bridging the gap to the "next world." So instantaneous communication didn't just have practical effects on society, but impacted the cultural imagination as well!

Just as a caveat, it's worth adding that the story about Rothschild and the Battle of Waterloo is not true – he does seem to have got information early, but not earlier than absolutely everybody else. The story is really part of an antisemitic myth, as I explored in an earlier response:

Did the red shield really defraud the UK after the battle of Waterloo?