How could I send a byte out of several Arduino MEGA uarts at the same time so that the data transmitted are all synchronized/in-phase? (i.e. if all are running at same baud rate I want the receiving peripherals to all get the byte of data at the same time)
I don’t think that is possible. The best you could do would be a phase delay between USART’s equivalent to the execution time of one single assembly instruction (62.5ns) and even this would require some clever low level programming.
If all peripherals will receive the same data always, you could wire TX of a single mega USART to the RX line of all peripherals.