How to avoid pulling the RX0 pin to reset sketch
It looks like your RFID readers are very chatty on startup, and the board is going into bootloader mode.

I'd consider either controlling the RFID reader's power using another pin (and a transistor or mosfet if the reader needs more than 20mA), or using a gate ( AND for instance, or a simple switch IC like the 4066) that you can control to disable the RX0 input until program start.

Another thing I've done in the past is multiplexed multiple serial devices to one serial input.  The RFID readers probably have outputs that indicate a card presence, so use those and a multiplexor.  When the card presence is indicated, connect that serial port using the multiplexor, read the card and act accordingly, then wait for another card detect event.