USB powered and battery powered on the same connector
If you are using a FT232R chip, this should work, provided you have a minimal voltage of 4.0V at the output of your LDO with all conditions. Other FT232 chips might have similar properties, check the datasheets.

The datasheet of FT232R states for VCC on page 11: +3.3V to +5.25V supply to the device core. However, in Note 1 (page 12), they precise that to use the internal oscillator (I suppose you want to save the space for a crystal) you need a minimum VCC of 4.0V.

Thus, the manufacturer ensures that the FT232R will reliably work, without an external crystal, with a VCC between 4.0V and 5.25V. This of course includes the USB communication with the USB host, which is of course an essential capability of the chip.

If you look at Chapter 6.2 of the datasheet (self-powered configuration), you can see that the self-powered circuit can be supplied with any voltage between 4V and 5.25V and still reliably communicate with the host. There's no reason this would not remain true if the power is derived from the USB supply voltage instead.

Although not mentioned in your question, another point you need to consider is the power supply voltage of the ATMega chip (I guess you will be using a ATMega168 or ATMega328, if you want to be compatible with Arduino Nano). Everything will be fine with your 6V external power supply as you will end up with 5.0V power. However, if you power from USB, the LDO will introduce a drop on the VBUS voltage, which is 5.0V nominal. If you use a 0.3V drop LDO, you will have 4.7V on the microcontroller, which is still fine. However, if VBUS goes below 4.80V, your VCC will drop under 4.50V, which is the minimum required voltage to operate at full speed (20 MHz), and the maximum frequency decreases with the voltage (see Chapter 33.3 of the datasheet for the ATMega 168P) In this case, you will have to operate at less than 20 MHz. If you use the 16 MHz typical of Arduino boards, you should always be in the safe area.