USB Communication between Arduino Nano and UHF Reader module
I've been under the assumption that USB operates using UART.


No. You have it backwards.  Devices can emulate a UART (known as CDC/ACM) over a USB connection as one of the many possible device profiles, but it is about as far from being a UART as an F1 car is from a Penny Farthing.

USB operates on a strict Master / Slave principle. That is, it has a single master device, called the Host, which coordinates all the devices on the bus and makes requests to them.  Everything you plug into a USB system is a Device and requires a Host to tell it what to do.

Neither the Nano, nor this adaptor you bought, are Hosts. They are just Devices. And Devices cannot talk to Devices. Only Hosts can talk to Devices.

Which means you need a Host in there somewhere. If you want it controlled by Arduino then you need a USB Host Shield (or something similar) which normally communicates over SPI.

If the device you are trying to talk to uses the CDC/ACM profile then you can use the USB Host Shield with it. If it uses a proprietary profile, like the CH340G or FT232, then getting it to work with the USB Host Shield will be much harder (maybe even impossible) unless you are intimate with the underlying protocol used.