We certainly know that since the fall of King's Landing, the siege of Storm's End was lifted by Ned Stark (evidence in following quote soon). Ned went to Storm's End after arguing with Robert Baratheon over the killing of Rhaegar's children. We can assume that news of the fall of King's Landing was flowing across the realm in the meantime (ravens, refugees, people, etc.). Certainly the new King would be informing Lords at castles, either to celebrate victory, or demand the bending of the knee.
The Tower of Joy is near Nightsong and Kingsgrave. Kingsgrave is the seat of House Manwoody. Nightsong is the seat of House Caron. Castles such as this would be the source of supplies, and news. As long as we assume the airspeed velocity of a laden raven is greater than that an army (or a band of 7 people on horseback), it is certainly possible for news to have reached the Tower of Joy with the events of King's Landing.
“I looked for you on the Trident,” Ned said to them.
“We were not there,” Ser Gerold answered.
“Woe to the Usurper if we had been,” added Ser Oswell.
“When King’s Landing fell, Ser Jaime slew your king with a golden sword, and I wondered where you were.”
“Far away,” Ser Gerold said, “or Aerys would yet sit the Iron Throne, and our false brother would burn in seven hells.”
“I came down on Storm’s End to lift the siege,” Ned told them, “and the Lords Tyrell and Redwyne dipped their banners, and all their knights bent the knee to pledge us fealty. I was certain you would be among them.”
”Our knees do not bend easily,” said Ser Arthur Dayne.
“Ser Willem Darry is fled to Dragonstone with your queen and Prince Viserys. I thought you might have sailed with him.”
“Ser Willem is a good man, and true,” said Ser Oswell.
“But not of the Kingsguard,” Ser Gerold pointed out. “The Kingsguard does not flee.”
“Then, or now,” said Ser Arthur. He donned his helm.
”We swore a vow,” explained old Ser Gerold.
Ned’s wraiths moved up beside him, shadow swords in their hands. They were seven against three. “And now it begins,” said Ser Arthur Dayne, the Sword of the Morning. He unsheathed Dawn and held it with both hands. The blade was pale as milkglass, alive with light.
“No,” said Ned with sadness in his voice. “Now it ends.”
The Kingsguard are clearly aware of the fall of King's Landing. Although to be fair, they only respond of that knowledge after Ned Stark's first comment regarding King's Landing's fall, and Jaime Lannister's betrayal. They don't betray any knowledge aside from that already mentioned by Ned Stark. So as far as actual proof whether Lyanna Stark knew of the specific events of Aegon's fate, is unclear.
That said, there is also no book evidence that Lyanna Stark named her son Aegon, or would have in any way done something with the knowledge of the death of Aegon Targaryen in King's Landing. It is entirely possible that a show writer figured Aegon was a cool name, and a cool reference to the removed storyline of Aegon Targaryen in the books, without thinking of the implications.
