How did Northrop Grumman propose to make the Global Hawk nuclear powered?
I didn't find any direct evidence that Northrop Grumman actually proposed a nuclear-powered Global Hawk. 

Citation 64 (2 April 2012) in the question doesn't reference a nuclear-powered Global Hawk, only "drones." The pictured Global Hawk was apparently just an illustration. 

Citation 65 (August 2003?) in the question does reference a nuclear-powered Global Hawk:


  The US Air Force Research Laboratory has funded at least two
  feasibility studies into nuclear powered versions of Global Hawk., [sic]
  which could extend the vehicles flight time from hours to months.


But this August 2003 page (above) appears to be copied from an earlier February 2003 article in New Scientist (below):


  The US Air Force Research Laboratory (AFRL) has funded at least two
  feasibility studies on nuclear-powered versions of the
  Northrop-Grumman Global Hawk UAV (pictured). The latest study,
  revealed earlier in February at an aerospace technology conference in
  Albuquerque, New Mexico, concluded that a nuclear engine could extend
  the UAV’s flight time from hours to months.


This is the only reference to a nuclear-powered Global Hawk I've found. I'm not sure this is a reliable source, however, for two reasons. First, this article was published in 2003, just five years after the Global Hawk started flying. That's not a lot of time for AFRL to conduct two feasibility studies for nuclear-powered Global Hawks, though not impossible. I've been unable to find these studies. It's possible these studies were for nuclear-powered ISR drones in general, not the Global Hawk specifically. Second, the New Scientist article references a problematic study by Carl Collins on using a metastable nuclear isomer of Hafnium (Hafnium-178m2) as a gamma/heat source by using xrays to induce the transition/gamma release. The claim has since been discredited (though this may not have been apparent at the time of publication). I'm not saying the author is unreliable (he seems like a decent STEM writer for the MIT Tech Review and Scientific American), but he doesn't seem to have a background in defense or aviation. So there's a possibility he conflated "Global Hawk" with "drone," which wouldn't be unusual of even good journalists then or now. 

Northrop's nuclear-powered UAV work seems to predate Global Hawk:


  Northrop Grumman is known to have patented a drone equipped with a
  helium-cooled nuclear reactor as long ago as 1986, and has previously
  worked on nuclear projects with the US air force research laboratory.
  Designs for nuclear-powered aircraft are known to go back as far as
  the 1950s. The Guardian


Feasibility of nuclear-powered UAVs

jCisco's arstechnica article gives a surprisingly good overview of nuclear-powered aviation (there's a couple photos I've never seen before). They touch on ANP, the nuclear Bounder hoax, Pluto/SLAM, and Putin's nuclear-powered cruise missiles and torpedoes. 

Some discussion of nuclear propulsion: Why don't aircraft use nuclear propulsion?. I'll repeat just one comment I made:


  @vsz the water boiler, turbines etc, have to be there and are heavy You
  can actually dispense with the steam turbines entirely. The HTRE
  series were basically modded J47s that sent air through a reactor
  instead of a combustion chamber. HTRE-3 pic. The whole engine+reactor
  assembly is actually lighter than a conventional engine+fuel,
  especially at long range. Rather, it's the shielding that's quite
  heavy. See SLAM/Project Pluto, the Mach 4 nuclear powered ramjet –
  Hephaestus Aetnaean Nov 28 '17 at 17:20


Northrop Grumman (maker of the Global Hawk) did sponsor Sandia to study the feasibility of using 'non-hydrocarbon' fuels for an "ultrapersistent" (read: nuclear) UAV. (Note that even this report doesn't mention the Global Hawk.) 2012 Report summary. Quotes:


  Upon completion of the CRADA, Sandia provided a final out-brief to
  NGIS UMS [Northrop Grumman Integrated Systems Unmanned Systems].
  Accomplishments: The effort concentrated on propulsion and power
  technologies that went well beyond existing hydrocarbon technologies.
  It contrasted and compared eight heat sources technologies, three
  power conversion, two dual cycle propulsion system configurations
  [indirect cycle?], and a single electrical power generation scheme [to
  power the ever-hungrier avionics mentioned earlier]. Overall 
  performance, specific power parameters, technical complexities,
  security, safety, and other operational features were successfully
  investigated. Large and medium sized UAV systems [probably Global Hawk
  or X-47B and larger] were envisioned and operational flight profiles
  were developed for each concept. Heat source creation and support
  challenges for domestic and expeditionary operations were considered.
  Fundamental cost driver analysis was also performed. System
  development plans were drafted in order to determine where the
  technological and programmatic critical paths lay. NGIS UMS and SNL
  felt that the technical goals for the project were accomplished. NGIS
  UMS was quite pleased with the results of analysis and design although
  it was disappointing to all that the political realities would not
  allow use of the results.


Taking this assessment at face value, nuclear-powered UAVs sound technically feasible.

The powerplant is well understood, suggesting a mature/pre-existing concept (eg a "conventional" nuclear fission reactor) rather than an exotic new concept:


  Basic application of the advanced propulsion and power approach is
  well understood


New power/propulsion concepts would probably fall outside the scope of this study, which sounds fairly small and limited to what the sponsor could potentially accomplish in the short/medium term. 



Note: "Conventional" is used in the same sense as the OP (I hope): "conventional reactor based on fissionable material." I do not use it here to refer to a specific reactor generation, fuel type, fuel phase, or coolant type.