ATMega328 and 7400 chips

TTL serial is active low. So if you used an OR gate, you'd get a high (idle line) if either of the inputs was idle, and gibberish if both were sending data.

You can or them with a pair of diodes (diode from RX of master to each slave TX) - you might also need a pullup resistor on the master RX; I don't think the AVR's enable the pullup on the UART RX pin) or an AND gate, but you'll get gibberish if both transmit at the same time.