I have been looking at the uart sockets on the side of the picocalc and I don’t really think they are useful without modification. The usbc connector is ordinarily routed to a usb-serial device (the CH340) and from there to UART0 on the pico. If you connect the usbc to a pc you will see the serial device appear when you turn the picocalc on. This is how the serial console works. If you attempt to use the UART0 sockets on the side connector then you will be fighting the CH340. This is unlikely to cause damage but it isn’t going to work properly. You can disconnect the CH340 from the pico with one of the internal switches and free up the UART0 sockets but doing so will also put the keyboard processor into its bootloader mode which is probably not what you wanted.
The UART_1 sockets are connected directly to the keyboard processor and will not be conflicted with the internal switch in its default position. I have no idea if you can read and write data on this port from the pico but if that is possible then it would work electrically.
The USB sockets go directly to the keyboard processor. Flipping the other internal switch will connect the usbc to these pins, presumably so that the firmware can be flashed. Again, I am not sure why you would want access to the usb pins on the keyboard processor in normal use.