The bombing of Hiroshima definitely did knock out communications, but not because of EMP (which is very short-ranged for weapons detonated in the atmosphere — smaller than the destructive effects — and anyway would have had much less of an effect on 1940s wired technology that it would have on modern circuits).

First, Hiroshima, like most cities, did have regular communication with the rest of the world. There was a large military base there. When communications abruptly end with things like that, you notice. And the Japanese military did notice very quickly that Hiroshima had suddenly gone quiet, and trains had stopped appearing from its rail lines. Note that by itself that would not necessarily signal an atomic bomb — it could have been some other kind of raid, for example.

Second, the Americans told them. Several hours after the bombing, the White House released a press statement saying that the city had been attacked with an atomic bomb. They also began broadcasting it to the Japanese islands via radio. (They would later also drop leaflets talking about it, but they weren't dropped until after the Nagasaki bombing.) The Japanese high command heard about these claims immediately.

Third, they sent a team to investigate. They by that time already knew Hiroshima had been devastated — nearby Army regiments had gone to the city and found it destroyed, and had begun the work of cleanup, helping survivors, and burial (the latter is one of the reasons there are very few photographs showing corpses in the city — the body cleanup operation began immediately). The Japanese high command sent a team of scientists, led by physicist Dr. Yoshio Nishina, to Hiroshima to confirm if it had really been a single fission bomb (as opposed to a large conventional bomb raid, which had become a regular occurrence at that stage of the war). Nishina arrived on August 8th, made observations about the character of the blast and the presence of radiation, and radioed back that evening:

What I've seen so far is unspeakable. Tens of thousands dead. Bodies piled up everywhere. Sick, wounded, naked people wandering around in a daze... Almost no buildings left standing. It's all true then? Hiroshima is completely wiped out? Completely. ... I'm very sorry to tell you this... the so-called new-type bomb is actually an atomic bomb.

So by August 8th/9th the Japanese high command knew that not only Hiroshima was destroyed, and that the US had claimed it was an atomic bomb, but had confirmation from their own scientists. (And the next morning, Nagasaki was similarly bombed. This is why I tend to emphasize how close the timing was between the two bombs — the Japanese high command was not really given adequate time to confirm what had happened at Hiroshima before they were hit again.)