Does the Widom-Larsen theory pose a credible underpinning for Low Energy Nuclear Reactions, often incorrectly referred to as "Cold Fusion"?
No, it doesn't.
"Heavy electrons", such as muons, can catalyze nuclear reactions by screening the charge. And electrons in solid-state crystals often have a high effective mass. So what can go wrong?
Nuclei have to get very close for fusion, far closer than the inter-atomic spacing. The electron needs to be become sharply localized in order to get enough probability in between the protons to screen them effectively. However, this means that there it loses the many-body lattice/phonon/etc interactions that increased it's effective mass int e first place.
First, "LENR" is not incorrectly referred to as Cold Fusion, it is exactly the same thing as Cold Fusion. The renaming is simply an attempt by Cold Fusion folks to politically distance themselves from Pons and Fleischmann, and people have a name for distancing the original discoverers--- it's called "plagiarism". It is fine to choose a name which is agnostic as to whether fusion is going on, but one must always say that this is what Pons and Fleischmann discovered: there is nuclear stuff going on in Pd/d systems.
Further it is very likely that fusion is exactly what is going on in this system, since this is the lowest energy nuclear process available in the system. He4 emissions have been reproducibly detected many times, starting with Paneth and Peters, continuing with Pons/Fleischmann, later McKubre, and most recently in Arata type cells. While these He nuclei could be alphas emitted from heavy nuclei decaying, it is much more likely, given the concurrent heat production, that the alphas are emitted by a process of conversion of deuterons, a cold fusion. All nuclear processes other than fusion of d and d, require more energy to overcome the Coulomb barrier than is conceivable in such a system. Even for d-d it requires heroic contortions, because it's three orders of magnitude in energy to overcome.
The Widom Larsen theory claims that the process going on in the cold fusion systems is that protons are absorbing electrons on the surface of the metal, where there is an inhomogenous electric field which can be very strong, and turning into neutrons. These neutrons then catalyze a bunch of energy releasing nuclear transformations. The mechanism is through the weak force, not the nuclear force.
It is preposterous, because there is a 1MeV mass difference between the proton and neutron, which is greater than the mass energy of the electron. It would be as reasonable to say that electron-positron pairs are produced at the surface of a metal. They aren't, and they can't be.
The strong electric fields at the surface of a metal are only so strong that they produce eV level energies for accelerating charged particles from the metal to infinity. There are no further energies available to bridge the gap to 1MeV of energy, and Widom and Larson simply make up this energy, by unphysical ansatzes for the near-surface field and made up energy balance requirements that pretend that MeV's of energy magically appear in individual particles on a metal surface.
Even if you admit that you made neutrons, we know what happens to metals under neutron bombardment--- the nuclei absorb the neutrons to get one unit heavier, the neutrons decay in a characteristic way in interstitials of the metal, leading to Wigner's disease, the neutrons escape in large numbers to trigger neutron detectors elsewhere (they are an uncharged and therefore highly penetrating form of radiation). They would not lead to any significant energy production (the same energy used to form the neutrons would be rereleased in their decay, aside from the comparatively little energy released in neutron absorption. He wouldn't be produced in any great quantities, although there would be some tritium).
Further, the formation of neutrons would catalyze transformations of +1 mass unit in the nuclei, leading Pd to transform to a heavier isotope, or into Silver through beta decay, if too many neutrons are present. This is categorically not observed in cold-fusion cells, what you see are a bunch of isotopes scattered at random in the periodic table in trace amounts, with all sorts of isotope ratios.
The theory does not work, it is energetically forbidden, and even if you put that aside, it does not explain the observed phenomenon--- heat production, He production, and elemental transmutations in Pd/d cells with a very wide spectrum of atomic weights.
The only reasonable explanation I have seen is the one I explained here: Why is cold fusion considered bogus? . The idea is that the KeV levels of inner shell electrons are exchanging energy with deuterium in the lattice, leading to fusion at 10KeV energies, which dumps its energy electrostatically into electrons or other nuclei, rather than by ejecting a neutron or proton. This process can explain most of the observations. It is possible that the deuterons form some sort of long-lived Rydberg-matter like state using the excitations deep inside the metal, or that they form metastable bands at KeV energies which cannot efficiently decay.
But independent of the true explanation, Widom-Larson doesn't make sense, because the theory is just a bunch of words strung together with no coherent relation to known weak interaction theory, or to energy conservation, or to surface theory of metals, or to known nuclear physics of neutrons. But one must commend Widom and Larson in one respect--- they were willing to believe the experimental data over their more powerful colleagues' head-in-the-sand denials. This makes it a little painful to criticize their theory, even though it is cockamamie nonsense.