An Arduino Uno has a second microcontroller on-board, an ATmega16U2. It is this chip you flash the MIDI firmware to. The Arduino Nano and Pro Mini do not have these chips, so you can't use them for USB MIDI (at least not without extra hardware). (The Pro Mini doesn't have a USB-to-Serial converter, the Nano has a single purpose one, not a programmable MCU.)
Read this: MIDI over USB · tttapa/MIDI_controller Wiki · GitHub
I'd recommend a Teensy or a (Pro) Micro.
Pieter