Risks of ftdi_eeprom? - TX always high after flashing
The ftdi_eeprom utility re-writes the entire EEPROM, not just the selected bits you have chosen.  That includes the "CBUS" pin configurations.

You need to check on the Nano's schematic which LED is connected to which CBUSx pin on the FTDI and set it accordingly.  You do that in your config using:

cbus0=TXLED
cbus1=RXLED


for example.

Possible settings for the CBUS pins are:


TXDEN
PWREN
RXLED
TXLED
TXRXLED
SLEEP
CLK48
CLK24
CLK12
CLK6
IO_MODE
BITBANG_WR
BITBANG_RD
SPECIAL


Except CBUS4 which only has the functions:


TXDEN
PWRON
RXLED
TXLED
TX_RX_LED
SLEEP
CLK48
CLK24
CLK12
CLK6


There is a chance that your pre-packages version of ftdi_eeprom doesn't support those settings, so you will need to download and compile the latest version from source from here: https://github.com/legege/libftdi

Also during testing I accidentally changed the VID of one of my boards thus rendering it unresponsive.  I did that by omitting the 0x from the VID and PID numbers.  Extreme care should be exercised using thin utility.  I have submitted a patch to allow the specification of a default VID for recovery from such situations.