This article will be a little shorter, as the 1600 turn signal circuit is a little simpler than the 1500 circuit (without the turn signal dual relay unit). If you think you may be missing some background here, you might review the prior article(s) on the MGA 1500 four way flashers.
For the MGA 1600 the flasher unit output goes to the turn signal switch rather than to the (1500 type) relay box, and the turn signal corner lights are powered directly from the turn signal switch with the front and rear lamps on each side permanently connected. Like my setup for the 1500 this schematic shows a single heavy duty flasher unit for simplicity, and a pair of diodes to operate the dash indicator lamp.

One significant difference with the 1600 is that the directional switch carries the full current of the flasher unit output rather than only the trigger current for the relay coils. When I install a double pole relay (or double pole switch) in parallel with the directional switch, the new relay (or switch) will also carry the full load of the flashing lamps. There are two lamps on each side, or three if you hook up a trailer. Load of three flashing bulbs will be 63 watts or 5-1/4 amps for each pole of the new relay. Since the lamps will be flashing when operated, the relay will be carrying this load intermittently with 50% to 75% duty cycle, so a 5 amp relay should suffice. If you were a little concerned about this you could use a 10 amp relay.

With any hazzard flasher setup the flashers have to work with the ignition switched off (or on). So the first step is to change the flasher unit supply wire from the fuse block A4 terminal to the A2 terminal to have always on power. Have fun determining the right green wire on the fuse block. The 1600 has only this one supply wire for the turn signals.

The next change is to move the two input and output wires from the original flasher unit to the new heavy duty flasher (which is not polarized). As with the 1500 setup you can leave the original flasher unit in place for appearance, but it will not be needed to carry the wire to the dash indicator lamp. If you remove the original flasher unit, tape up the loose wire end (for the dash indicator lamp) and secure it to the side of the harness.

Next under the dash it's time to install the new double pole relay (or switch). The two relay contact poles are used to connect the directional switch input terminal to both output terminals on the directional switch to have all four corner lights operate together. These three wires are connected the same way as on the 1500, except do not connect the flasher output to the new relay trigger coil.

The new heavy duty flasher most likely will not have a pilot lamp output terminal. To compensate for this you can add two diodes (cheap from RadioShack). Connect the diodes to the left and right turn signal lamp circuits. The easiest way to do this will be to connect the diodes on the new double pole relay (or double pole switch) output terminals. Then connect the other end of the diodes together and connect that to the dash indicator lamp, which happens to be right next to the directional switch. If you do not want to cut and splice the original harness wire for the dash lamp, you can purchase a new bulb holder for the dash lamp and tape the original bulb holder to the side of the harness for safe keeping.

Power for the new relay trigger coil must come from an always hot source, and the only always hot wires under the dash are at the ignition switch input terminal or at the lighting switch input terminal. If you install a barrier strip for the wires around the ignition switch (as shown in the 1500 article) you can connect the new relay trigger power wire there. The other terminal on the new relay for the trigger coil will be connected through a switch to ground. As noted in the prior article for the 1500, you can use the original fog light switch if your car does not have fog lights. Otherwise you would need to install another single pole switch to operate the new relay, which sort of defeats the reason for using a relay. Instead you could use a manual double pole switch in place of the relay.

You can secure the diodes to the new relay (or switch), and you can secure the new relay to the turn signal switch (as shown in the article for the 1500 flasher setup). As noted in other articles, the heavy duty flasher unit will defeat the function of no-flash if a corner lamp bulb is burned out or disconnected. So make a note to yourself to be diligent at checking the corner lamps regularly to be sure they always work.

A more exotic version of the four way flasher installation would call for using two flasher units so the original flasher would be retained for its non-flash function when a bulb is burned out. To do that you need to have another pole in the relay (or manual switch) to disconnect the original flasher when the heavy duty flasher is connected. You could use a three pole double throw switch, or a two pole switch and a two pole relay. This is all jolly good fun to pick your favorite way to add complexity, but don't make it too complex, and do make it robust to retain reliability.