Update on this issue:
For players on Mobile platforms, the free Minecraft music pack will need to be downloaded from the Marketplace in order to be used, and for jukeboxes/music discs to be played.
I don't know why this happens, but music discs don't work for me!
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Moved to the correct project. Please update the platform field.

Yeah, they don't seem to play even for me. I have tried EVERY SINGLE DISC, and EVERY SINGLE GAMEMODE, and there's NO MUSIC. I have the device volume turned up to the highest I can get, and I do NOT hear a thing from music discs. All other sounds (eg. walking, destroying a block, etc.) work just fine.

Try downloading the free music pack from the store
You must download the Free Music pack from the marketplace, otherwise it won't work.
Thank you for your report!
However, this issue has been temporarily closed as Awaiting Response
Is this still an issue in the latest version? Could you please confirm that you have downloaded the free music pack from the Marketplace, and that your volume is set sufficiently?
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@[Minecraft] Mega_Spud (Jay Wells) but why we have to download a music pack? Is not possible to implement the music in the game?

The music files are relatively large. Some people incur network charges when they download a release. Placing the music files in a separate download gives people a choice whether they want to spend their minutes on downloading something that isn't necessary and often isn't used at all while playing.

Hello, I wants to download the music pack, but it remains stuck importing. I tried with Wi-Fi and phone data and nothing. I also relaunched the application but still nothing
[media]
Music pack was deleted