The whole thing (inc. 120led 2metre strip) is getting power purely from the 5v USB input on the Nano. From your reply it seems I have indeed been duped by Youtube tutorials. That was where I got the idea and the lights worked fine for a week or so before dying. So, should I continue with the circuit I have and simply add an extra 5v supply to the LED strip or should the LEDs be on a completely separate power circuit with only the data line connected to the Nano?
OK, you might just clarify a few things here.
What is the current state of the system? Is it working when it cools down? If not, can you actually program the Nano when disconnected from the LED strip, such as to run the "blink" test? If that fails, you will presumably need a replacement Nano (clone).
Now, a 120 LED strip would draw about 6.5 Amps at full white brightness, so to be safe, you want a regulated 5 V power supply capable of providing that (and connecting that, always both 5 V and ground, to both ends of the strip and preferably the middle as well), so a 10 Amp nominal supply would be appropriate. As explained in #3, this supply not only can but should be used to power the Nano with the three wires from the start of the LED strip where you connect the power, back to the Nano - ground, data and 5 V to the "5V" pin.
We are curious as to what supply you have been using so far.
You should also have a 470 µF or so capacitor between 5 V and ground where the data feeds the strip (if the strip has connectors attached, it usually has two sets of wires, such as a white and red for connecting the power and the capacitor, and white, red and green with a plug for connecting the controller; the colours may differ) and a 330 or 470 Ohm resistor in series with the data wire.
Back to you!