How would one decide, if making this from scratch, to use 4kΩ, 1.6kΩ,
and 130Ω resistors?
The resistance values determine how much current can flow in the associated parts of the gate circuit. This then determines the input and output currents and operating speed of the gate.
R1 limits the current sourced by an input when pulled low. The previous gate ...
If all UART comms protocol originates at the Port of the MCU then this may be a workable solution to MUX to the various target devices.
On the other hand if UART comms can asynchronously originate from any of the target devices at any time then this multiplexing scheme is a non-starter.
I'm not sure how big each UART is (and the necessary multiplexers).
Some time ago I had the intention to use several UARTs (like 4-6) for MIDI. My idea was to use STM32F103C8T6, which have 3 UARTs (each). Using such MCU, you can easily gather all UART info and send it (e.g. via SPI) to your LPC MCU.
This also prevents the problem related to buffer/timing ...