It's not exactly as you describe - in fact Marconi's signal is the first message that is replayed - but this has significant similarities with "The Waveries" (1945) by Frederic Brown, first published in Astounding, January 1945.
Reception of old signals starts without warning:
"—only the best tobaccos," said the radio, "go dit-dit-dit—the nation's favorite cigarettes—"
George glanced at the radio. "Marconi," he said.
He meant Morse, naturally, but the whiskey sours had muddled him a bit, so he was nearer right than most people who heard that dit-dit-dit. It was Marconi, in a way; in, as it turned out, a very peculiar way.
Stations on Earth stop transmitting:
Casey Blair said, “It is now three o'clock in the morning. The radio interference has gone on for five hours and has reached the point where not a single current program is getting through. Every major broadcasting station in the world has gone off the air.
A scientist explains the replay of all broadcasts:
"Why not, sir? They are creatures of ether, not of matter. Ether permeates space uniformly. They were, until they were attracted here, at a point in space not greater than twenty-three light-years away. Our first indication of their arrival—rather, the arrival of the first ones, if you want to put it that way—came with a repetition of Marconi’s S-S-S transatlantic broadcast of forty-six years ago. Apparently that was the first Earth broadcast of sufficient power to send signals which they could perceive at that distance. They started for Earth then, presumably. It took twenty-three years for those waves to reach them and twenty-three years for them to reach us. The first to arrive had formed themselves, imitatively, to duplicate the shape, as it were, of the signals that attracted them. Later arrivals were in the form of other waves that they had met, or passed, or absorbed, on their way to Earth. There are now fragments of programs which were broadcast as recently as a few days ago... uh... wandering about the ether. Undoubtedly also there are fragments of the very last programs to be broadcast, but they have not yet been identified."
Things go downhill from there, with all electrical devices stopping, and eventually the world running on steam- and muscle-power.
You can read the story at the Internet Archive.