SD Card not readable after upgrading to El Capitan
I would install the older OS to another USB drive so that you can verify that the new OS is really causing the issue and it's not a hardware failure.

When you boot to the old OS (hold option key when booting to choose which OS of the ones connected) run diskutil list to check the filesystem on the SD card.

It's unlikely that it's a format that's no longer included in the new OS. It's more likely that you'll find the card itself faulty or the hardware has changed and needs to be cleaned (lint can obscure one pin connector) or repaired. You can also get a USB to SD card reader to avoid needing to reinstall the old OS just to eliminate the SD card reader as a potential source of the problems on the Mac with the new OS.