How can I increase the number of hardware UARTs in a design with a single UART MCU?
You could add an extra UART over SPI or I2C. These have internal FIFO buffers inside them so you periodically check them for content and let them handle some buffering. NXP, TI, Maxim probably offer these parts. MAX3100, SC16IS752 etc. You could also add a second MCU to do this, but as you're already using quite a simple microcontroller with 1 UART and want to stick to that, that seems overkill.

You could also use a multiplexer to choose between the UART your micro talks to. But as you mentoined data is transmitted simultaneously , that won't work.

Software serial could be your last option but it can be very annoying to deal with if your microcontroller is doing other stuff as well.