Arduino Nano + RFID RC522 + Ethernet shield W5100
I had almost exactly this situation with one of my projects.

The problem is that some Serial Peripheral Interface (SPI) devices don't properly put the MISO pin into high-impedance when SS is pulled high. I eventually resolved this by placing a tri-state buffer in the MISO line for each of the SPI devices.

I used a 74HC125 (4-channel tri-state buffer).

This is the page on Better SPI Bus Design in 3 steps that finally got me on the right track. I think the explanations are reasonably clear (at least compared with many other sites I found!).