Could this work?
Sure! We already have rocket-propelled torpedoes that make use of sea water as part of their propulsive reaction mass:
Once accelerated, speed is maintained by an underwater ramjet fueled by hydroreactive metals using seawater as both reactant and the source of oxidizer; the torpedo travels at around 200 kn (370 km/h; 230 mph).
Clearly overpowered jet systems can work!
Would it be efficient?
That's very difficult to establish, because fluid dynamics is hard and I'm not about to calculate the drag force on a supercavitating submarine for you. Remember though that the drag force scales according to the square of the velocity, so if you wanted efficiency you'd be moving slowly rather than expending all your energy ramming seawater out of your way.
ocean being polluted?
The ocean will inevitably become polluted, because you're exhausting very hot nuclear coolant directly into the environment, and that coolant will contain at the very least all sorts of interesting neutron activation products from the dissolved materials in seawater, notably sodium-24 and chlorine-38 which will at least be thoroughly diluted and have relatively short half-lives.
Are there any unintended consequences
The biggest problem that solid-core nuclear thermal rockets are likely to face is buildup of neutron poisons in the reactor core which will reduce reactor power output to the point where it eventually becomes useless. Project rho has a few things to say on the subject, but doesn't get into the gory details.
A clever nuclear reactor design can work around this to some degree and be factory sealed and still operate for years... apparently some US navy reactors are like this, and the ships they're mounted in will be end-of-life before they run out of fuel. Your ridiculous nuclear rocket on the other hand is operating at a much higher power level, which means the fissiles in the reactor core will be used up more quickly and the neutron poisons will build up faster.
This means you'll have to refuel more often than a regular nuclear reactor, and you'll either need a good source of highly enriched fissiles, or a nuclear fuel reprocessing facility. The former is obviously a proliferation risk, and the latter will also be a source of plutonium as well as clean fuel rods.
Such facilities are hard to hide, even without you running a giant underwater rocket that you could hear from the other side of the world and leaves a surface-visible wake of warm radioactive bubbling seawater. This makes you a global nuclear threat, and inevitably there will be some kind of consequence unless you're a powerful nation-state.