The W10M app store is still operating, and the Lumia 1520 runs W10M just fine, so one option would be to upgrade the OS. You might need to side-load (using the developer tools) an app to "trick" the update process into thinking your phone is a different one so that Microsoft will send it the update, or at least use the Windows Insider app (which might not be available or even work if it was anymore).
Another option is to use the "WPInternals" tool to unlock the bootloader of the phone, which allows configuring the phone for USB Mass Storage mode. In this mode, you can browse the phone as though it's a flash drive (the phone OS isn't running at that time), which would allow you to image the old phone (or just extract specific info) and copy it to the new phone. Be aware that some parts of the phone's storage are unique to each device, and if moved between devices will make things like the cellular radio stop working!
Note that if what you want is the apps themselves, rather than the data stored within the apps, it is rather difficult and probably not worth the effort to extract them from the old phone. W10M is backward compatible with apps all the way back to WP7 - I sometimes play Puzzle Quest 2 or use the original OneBusAway app on mine, those came out even before WP7.5 - so unless the apps were pulled from the store (which sadly does happen sometimes) you can just install them on any phone running W10M. If you do need to extract the apps, that's technically possible (with some hacks, requiring unlocking the phone for file-system access in one of several ways) but it's not easy, and the apps will be considered sideloaded (rather than installed from the store) which means no updates and may limit the number you can install.
However, if you want to get the data from those apps (such as saved games, app-specific media or documents, personal customizations, or so on), you may need a way to copy them off the old phone. Apps written for WP8.1 or later can sync some of that data automatically through Microsoft's servers, but there are limits and most of them don't even take full advantage of that feature.
As a side note, even if you could move all your apps onto the SD card and then took that SD card to a new phone, it wouldn't be able to use the apps (or their data) on the SD card. Partially because the apps wouldn't actually be installed - just like on desktop Windows, installing an app does more than just copy files to the install directory, it also changes the registry - and for mobile apps those registry changes that happen at installation are required before it's possible to even launch the app. The bigger problem is that all app installations and data files stored on the SD card are encrypted, using an encryption key that is unique not only to each phone but also to the "OS installation" on the phone. Even just doing a hard reset is enough to make the phone unable to read the app data on the SD card (although other stuff on the SD card, like documents and pictures, remain readable by anything).