Several things could be going on.
1.) Not enough current for the SD card reader to boot up/work perhaps?
2.) How are things wired? Do the RFID reader and the SD card reader have different CS pins? Are you toggling them correctly? (when you need to use/access either of the devices?
3.) Is the Nano or a +3.3v device? or a +5v device?
4.) If a +5v device... is this SD card reader a shield? Where does the level/logic shifting come into play here?