The problem isn't the reactor, it's the shielding. The US had a huge aircraft nuclear propulsion project around this time, although they never ended up flying anything. ....
Understand, I'm not talking about fission or radioisotope power for
flight; I'm talking about a power source for motion while submerged--and while we are at it, we need power for the "hotel" load and basic operations while below as well; compared to the power needed for a 5 knot creep this is probably the major power draw. Something like 1 percent of the sort of power output needed to propel the airplane while airborne!
If one did have a fission or radioisotope source that could keep a plane airborne, then it would probably last something like a year at full power. The thing to do then is simply keep the plane flying and rely on its mobility for defense. The whole question of submerging to loiter would not come up at all.
I've seen a RAND Corporation study of big airplanes (to nominally replace the C-5, so the target payloads were something like if a C-5 were scaled up to mass 1000 tons) that considered various possible alternative fuels--liquid hydrogen, liquid methane, even ammonia--and compared the planes needed to match a baseline plane based on conventional jet fuel (which, for purposes of economic comparison of the costs of the other fuels, was assumed to be synthetic--this study was from the late 1970s or early '80s and was considering the future in the light of exhaustion or strategic denial of petroleum). And included in the mix was a possible nuclear fission option--the power source would be a sealed liquid-metal cooled reactor unit, one that would feed a secondary loop of molten sodium-potassium mix (pumped electromagnetically) to a heat exchanger that would be basically equivalent to the combustion chamber of a conventional turbofan. (Unlike the conventional locations of the turbofans in pods below the wings on the other versions, the fission-heated jets were located in a cluster right above the power plant which was installed at the wing/fuselage junction for reasons of mass balance, thus dividing the volume of the fuselage in two, forward and aft of the nuclear installation). As a concession to safety, the plane would have to take off and land with the fission core shut down, burning conventional (synthesized) jet fuel in the engines instead to serve as conventional jet engines. Upon reaching cruise speed and altitude, at a good safe distance from the probably inhabited airport region (and avoiding other populated areas, ideally flying mainly over the ocean) the nuke plant is activated and the plane switches over to nuclear operations. Before landing, the plant is shut down and the engines are again switched over to jet fuel. Also there was a reserve in case the nuclear plant had to be shut down for some other reason while airborne.
This mode of operations did tend to rob the nuclear plane of much of its advantage as the jet fuel needed for these auxiliary but vital operations was substantial; the silver lining was that the fuel reserves could serve as shielding! The nuclear version was the heaviest alternative by far; the lightest though bulkiest was the hydrogen-fueled one.
Anyway had it been possible to actually make such a plane, the study pointed out the radical options its year-long endurance could enable, as a recon plane, command platform, or missile carrier. And while using it for the nominal design purpose as a cargo transporter would involve landing and taking off twice (thus using up the fuel reserves and requiring more--plus the study did not discuss the typical problem fission plants have that when they shut down they "poison" themselves with decay isotopes that absorb neutrons and so can't be restarted until these decay, a process that takes days--perhaps certain fission reactor core designs can avoid or minimize this, allowing the plant to be restarted mere hours after being shut down?) at least range is obviously no problem--once airborne, this thing can haul your cargo as far as you like, even if the world were a lot bigger than Earth!
I wish I could look that volume up again but it was at a university library in another state from where I live now.
I have thought up an alternative mission, analogous to attack subs versus "boomer" strategic missile subs--the plane could be meant to attack ships. In that sense it is more like conventional submarines, but with range and attack options like an attack plane so it is also an alternative to having aircraft carriers. Being submersible not only helps avoid detection until the attack is underway, it also enables the plane to stay on station despite the possibility of storms that might sink a surface-parked plane, by being submerged safely below the waves of the storm.
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To get away from the nuke tangent, I tried a couple Google searches for using sodium and water as a power source but have not yet come up with anything but fuel cell proposals. Of course a good fuel cell design that can use seawater as a reactant would be perfectly good, even better actually than using the reaction as a heat source!
A magic word seems to be "pyrophoricity." Anyway, I have yet to come with anyone proposing to fuel a submerged vessel with any kind of metal/water reaction, for heat or for direct electric current. Searches on "sodium submarine" mostly come up with either references to the
USS Seawolf (the second US nuclear powered sub, that used a sodium-cooled low-pressure reactor as an alternative to Rickover's now-conventional pressurized water reactor--it was not very successful and its reactor was eventually replaced with a pressurized-water one) or to the salt content of sandwiches!
Oh, and
this.
Is that helpful?

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Well, damn. On the second page of the search, there was
this! Still, since this is for a fuel cell that again uses stuff onboard (that happens to include sodium) and does not use seawater as a reactant, it's conceptually the same as any other battery or fuel cell concept. I suppose if I keep digging I might find that someone somewhere sometime has proposed something more serious than a bathtub model submarine based on baking soda bubbling, but the more of these kinds of things I find the stronger my conviction that the notion has been considered--and rejected, in favor of alternatives we already know are limited.
I'm down to page 10 of that search--still the same mix of baking-soda bubbling toy subs (but one from the Clabber Girl site, so that's authorative!

) articles on the Seawolf, stuff about submarine sandwiches and health, and battery/fuel cell systems, with miscellaneous stuff about the chemistry of the Dead Sea and so on.
Lotsa luck, y'all! I'm assuming any submersible plane designed in the 1950s or after is going with some kind of small nuclear plant for submerged power, or plans to surface occasionally to recharge batteries a la conventional diesel sub operations. (A bad idea for a strategic boomer but possibly OK for maintaining an attack-plane fleet in being on dispersed stations).
Just a hundred or so horsepower should be more than enough for slow submerged motion; you need more than that to keep the lights on, the air purifiers flowing, the controls operational, etc...