Mux UART with 74xx Multiplexer
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 issues for switching between the UARTs. Note there also STM32 MCUs which have 8 UARTs (some STM32F7 models at least).