My recommendation would be to do a hard reset (Settings | About | Reset my phone). You'll want to make sure your photos are all backed up to either SkyDrive or your PC first, and you're going to lose app data for apps that don't provide a backup method (SkyDrive, app's servers, etc). If you have the backup settings on then your apps will be able to automatically reinstall when you restore the backup. Also, if you have GDR3 installed you can do that restore from WiFi.
If after a hard reset the buttons still don't work, the problem is hardware. If it is still under warranty, contact Nokia for repair. If not, hopefully you've got insurance on the phone. Also, there are plenty of indie repair providers out there - you could check with them for a cost estimate, should you not be covered by insurance.
Check [https://guideusermanual.com/product-name-lumia-1520-at-amp-t-manual&po=672216&lang=English|Nokia Lumia 1520 Manual] for your more assistant support.