I found an article where the author collected a few songs which people listen in '56 or were inspired by the events of the Revolution.

Ahogy lesz, úgy lesz
It's cover of Whatever Will Be, Will Be by Ray Evans. After the Revolution it was basically banned.
https://youtube.com/watch?v=7vL6igSQxPU

Honvágy
Another cover: Memories Are Made of This - a hit of Dean Martin in 1956.
https://youtube.com/watch?v=5ruBE72iD8o

Avanti Ragazzi di Buda
A march of the Italian football club Lazio, starts with a nod to the Revolution. Very political song.
https://youtube.com/watch?v=dbp4bjaiwjo

See You Later Alligator by Bill Haley, was broadcasted frequently on Radio Free Europe.
https://youtube.com/watch?v=Ch1UQ47rWKU

Peace in the valley by Elvis. This particular performance was dedicated to the victims.
Ed Sulliven in his intro gives the background what's going on:
https://youtu.be/3XYm5iECuTI?t=766
(It starts at 12:46 the link points to that point.)
Some additional info in this video's description:
https://youtube.com/watch?v=zJARM4r_2E0

Egmont by Beethoven
During the night of 23rd some revolutionaries acquired a mobile radio station (a truck), they parked it next the the building of the Parliament. Also obtained a couple of vinyls with the Egmont among them. They played it a lot.
https://youtube.com/watch?v=554ZJUv-avQ

Radio Free Europe of 1956. Right now it is being streamed "live" (from 19th October to 12th November). Lots of Hungarian talk (maybe in Russian in the background), should be some music too. The record was stored by the National Széchényi Library and digitized not long ago.
As I listen this it's news from Poland.
https://youtube.com/watch?v=E0ZQFp8gEE8