The Nano was designed and genuine boards are produced by Gravitech. They have published the schematic of the V3 board on their website. Unfortunately, it has an error on it. I believe genuine Gravitech boards are fixed, but the many chinese copies simply implement the published design, complete with bug. If you want reliable USB on a cheap Chinese Nano, you probably need to do some soldering on the bottom of the board.
The nano uses a DS_FT232RL for usb (chip on the bottom). If you look at the data sheet you will find that pin 26 is 'Test' - "Puts the device into IC test mode. Must be tied to GND for normal operation, otherwise the device will appear to fail." I expect you might see where this is heading already...
If you look at the nano schematic pin 26 on the FT232RL is ... disconnected.
However, Pin 25 is connected to ground on the nano. So a solder bridge between pin 25 and pin 26 will the problem. You want a solder blob between pins 25 and 26, being the fourth and third down on the right hand side when you look at the chip with the dot at top left. Obviously, you don't want to short the adjacent pins into this blob.
Alternatively, if you look further up the board you can see a reasonable expanse of ground plane in the traces, and notice that the top end of the right SMD component just above the chip is connected to that. You could put a flying wire from pin 26 to the top end of that component. I think the solder bridge between legs is easier.
With blob added, the Nano communicates reliably over USB. Without it, it's a lottery.
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