How do I multiplex 6 x 4 bit ALU outputs?
You can use 6 - 4 bit tri-state buffers - like the 74HC125

each of your ALU outputs goes into a separate 4 bit - tri-state buffer - with the buffer enables commoned-up on each group of 4 bits. The outputs of the tristate buffers can simply be wired together onto a 'bus' and you then use the 6 buffer enable lines to select which of the ALU outputs is selected onto the bus.

you could also add a 74HC138 to convert a 3 bit selection into the 6 buffer enable lines