Assuming a planet with (water) oceans and technology similar to that on (current) earth, but instead of storing the radioactive waste in caves, they just dumped it into the ocean (in strong, heavy containers such that it does not leak) - what would be the environmental impact?
Assuming containers that don't leak? Not much, because water is an excellent radiation shield. If you just piled the containers up (carefully, so as to avoid accidentally assembling a critical mass), you'd get a dead zone extending a few meters outwards from the pile. Over time, a dead-but-not-decomposing pile of sea life would build up in and around the dump site, eventually burying it and keeping the pile from growing further. Outside of the dead zone, the rest of the ocean will continue on unconcerned.
The trick is making containers that don't leak. The ocean is a rather hostile environment for most materials.
It looks exactly like our society, because we've done this. It was one of the things Greenpeace were heavily against in the 80s and 90s.
There are also a number of sunken nuclear submarines, and tons of contaminated water from Fukushima.
Dilution is very powerful and the ocean is very large. The nearest this might come to a risk is the biological re-concentration of certain materials (Iodine, cesium) through the food chain.
One method that has been considered for disposal of nuclear waste is to put it in containers and drop it into deep ocean mud. The objective is to get the container to sink into the mud, once it's buried it doesn't matter if the container fails. The impact is basically zero.
Even if you don't bury the containers the effect is minimal. It's hundreds of years before the water comes up to the surface, during that time most of the radioactivity will have decayed (especially if you remove the useful stuff first.)
1) This is too much for a comment, but it sets you on your way
2) I am not going to do all this homework, that takes hours]
Assuming that the containers will break down you will have to make believable estimates of:
1) Half-life of radioactive isotopes considered dangerous to life.
2) Average time it takes the containers to break down and isotopes to seep out (and it makes a whole lot of difference if these are e.g. contained in glass-like materials).
3) The time it will take for these isotopes to ascend from the deep sea to shallow waters where they will have an impact on life - most importantly on our sea food, with its accumulation effects.
4) Probable health effects of continuous exposure to low levels of radiation.
Points 1) and 2) can be found with some research. 3) is more difficult.
The speed of vertical convection of the water mass is hard to guess.
Quoting from Chapter 8 - Ocean circulation, of "Introduction to Ocean Sciences" by Douglas A. Segar:
Thermohaline circulation is difficult to study, and most of our knowledge of it comes from studies of density and other characteristics of the deep-ocean water masses. Much of our understanding of thermohaline circulation comes from modeling studies, but the models are themselves limited by the relatively small amount of data that is available to calibrate and test them
Read that publication from the section 'Thermohaline circulation' at page 190 to get a general idea and make your estimates.
For point 4) we do have some information from nuclear accidents, atomic bomb blasts and industry workers; you would have to match their exposure doses to the ones you estimate coming out of your back-of-th-envelope calculations for 1) - 3)
And I forgot the obvious 5) Amount of material that was dumped
Maybe they will not get problems with global warming and oil wars and have enough power to make as much drinking water as they need.
There are few down sides if the containers do not leak EVER, but if containers are found to be leaking it is VERY hard to do anything about them.
Dumping into the ocean is one of the safest methods
As much as some like to try to portray nuclear waste as a sort of radioactive Bogey Man, the reality of it is that nuclear waste is boringly easy to deal with. This comes from the fact that the substances that necessitates long term storage — Plutonium, Americium and the other Actinides — have a perculiar chemical quality:
All the long term nuclear waste loves rock and mud
Stick nuclear waste in mud and it remains there.
And here is the thing that is so neat about oceans:
There is a lot of mud in the oceans
So if we were to completely ignore the public relations nightmare that stems from the words "Hey, let us chuck our nuclear waste in the sea", this is actually an extremely effective and safe method of storing nuclear waste. You find a big geologically inactive mud flat in and ocean, put the containers on that, and let them slowly sink down into the mud. And that is all she wrote.