I want to add an under-counter auxiliary water heater to my 15A dishwasher circuit. I need to install some sort of relay to cut out the water heater when the dishwasher is being run, to avoid overloading that circuit. What kind of relay do i need for this? Both appliances have regular 3prong 15A plugs.

If I understand correctly you have 15 amp breaker with 14gauge wire.
Total watt capacity of 15 amp breaker is recommended at 80% rated load = 12 amps

You want to connect two appliances to same 15 amp line.
1) You have 6 amp Dishwasher that draws 720 Watts
2) You have under the counter water heater with 1500 Watt element that draws 12.5 amps.
3) Combined load is 18.5 Amps and 2220 Watts, approximately
So you do not want dishwasher and water heater turned on at same time.
Makes good sense.

Solutions:
> Easiest thing would be to put water heater on ordinary wall switch located inside sink cabinet.
Back when garbage disposals first came out, garbage disposal switch was inside sink cabinet. You opened cabine door and flipped the switch.
> Next solution would be to put water heater on a timer located under the sink. There are different timers. In-wall timers and box-type

In-wall timer would need relay/contactor
A battery operated timer like the ST01C would not need to be reset after power outage
http://waterheatertimer.org/Programmable.html
Here are instruction pages for selecting and installing the relay/contactor.
http://waterheatertimer.org/H230A.html
http://waterheatertimer.org/Install.html
Add a comment any time.

Box-type timer like T101 120Volt timer. This timer needs no relay/contactor.
http://waterheatertimer.org/How-to-wire-T104-Intermatic-timer.html#T101
Timer needs to be reset after power outage, but operates on ordinary 120V power, and is simple to install.

> There is no easy way to wire the dishwasher and water heater together so one is certain not to operate when other is turned on. There are likely electronic control panels for that project. But since you know the hours when dishwasher is likely to be ON, then set water heater timer to turn OFF during those hours. Water inside tank will stay hot for a couple hours, so there is no loss.

It sounds like you are a solid electrician.
If needed, add comment any time for more assistance.

Why not stop at hardware, or electrical store and ask for a single pole, double throw switch. This will allow you to do want you want, without any relays etc. When in position 1 for example, power would flow to dishwasher only,and none to water heater. In position 2, power would flow to water heater only, and none to dishwasher. Simply mount it in a convenient spot where wires can reach, and so can you.