How to detect if LED is switched on in external circuit using Arduino NANO?
I'm afraid your circuit does not work.
In fact, the base of the transistor is always connected to the power, regardless the LED being ON or OFF. Therefore the BJT would be always in the ON-state. Also a current limiting resistor in series to the LED is missing (but I think that it was just for sake of simplicity).

Assuming that:


you can mount in series something to the LED
an additional 1.3-1.5V voltage drop will not cause too much issues (to be short: if the external circuit power supply is "large enough", and if you will not be too much upset if the LED luminosity will be smaller).
You want to sense a signal LED, i.e. let's say with a current smaller than 20 mA.


Then, the safest way would be to use an optocoupler, such as 4N25-4N35. Put the LED of the optocoupler (transmitter side) in series to the LED you want to "sense", and use the phototransistor side to sense the LED status: it's emitter goes to the GND to your Arduino Nano, the collector to any input, with pull-up (The pull-up is mandatory!). 

In this way, when current flows across the LED, it will also flow in the optocoupler LED, therefore the optocoupler phototransistor will be ON.

Also, using the optocoupler the two systems are electrically insulated each other (and you won't need to share the grounds).

Below you can find an example.



EDIT:
If you cannot wire the optocoupler LED in series (e.g. not enough voltage headroom because the power supply is too close to the LED Vf, or your LED is already on a PCB), you can connect the optocoupler in parallel. 
Assuming the LED current is much larger than 1mA, you can use the following circuit. Choose RO so that the current flowing on the optocoupler will be 1mA (390-470 Ohm in your case). Usage of high current-transfer-ratio (CTR) optocoupler is suggested, but this might work also with a 4N25 or 4N35 (you'll be pretty close to the limit...)