Assuming the theories that the Gluon field and the Higgs field are both responsible for particles gaining mass, how would both exist simultaneously in the same world? How would they interact with each other? And why isn't one field alone responsible for particles gaining mass?
Edit: Found this link to a 2013 question that appears to answer this one...
There is no theory that suggests that gluons generate mass (at least, not in the sense of the Higgs field). Gluons, as any other kind of particle, are responsible for the "kinetic mass" of objects, due to $m^2=p^2$. But they do not generate mass through spontaneously symmetry breaking, as the Higgs field does. These particles are very different, and their contribution to the mass of the rest of particles have nothing to do with each other; it is an entirely different mechanism.
Gluons and Higgses don't interact directly, but they do through loops (which is in general non-negligible).
The Higgs field does not couple to the QCD gauge bosons or gluons in the same way they do with the the $Z$ and $W^\pm$ fields of the weak interaction. The breaking of EW symmetry means the massless gauge bosons of weak interaction, massless meaning their degrees of freedom are transverse, absorb Goldstone bosons of thHiggs field and those degrees of freedom become longitudinal modes of the bosons. Elementary QFT illustrates fields with longitudinal modes are massive. Gluons remain massless.
For the QCD group $G~=~SU(3)$ and a subgroup of it $K$ to which it is spontaneously broken, the broken generators construct the coset space $H = G/K$. The broken generators are the axial symmetries in the chiral symmetry of low energy QCD, $SU(2)\times SU(2)/SU(2)_{isospin}$ in a broken symmetry, with the unbroken ones being isospin. Think of this as moding out the unbroken symmetry from QCD to leave the broken axial symmetry behind.
This low energy QCD theory, which produces the theory of baryons and mesons is a sort of Goldstone process. This looks remarkably similar to the Higgs mechanism. In fact the quartic potential of the Higgs field appears suspiciously similar to the Lagrangian for a nonabelian Yang-Mills gauge field. This prompted Hitchens to advance the so called Higgs bundle, which consists of a Yang Mills principle bundle plus a dual scalar field.
Assuming the theories that the Gluon field and the Higgs field are both responsible for particles gaining mass,
They are not. The Higgs field is responsible for the mass of elementary particles in the standard model of particle physics. The particles with zero mass, like the photon and the gluon do not interact directly with the Higgs field .
how would both exist simultaneously in the same world?
Easily, since they do not interact
How would they interact with each other?
In Feynman diagrams containing gluons which go into virtual loops with fermions a higher order interaction can be written down.
The basic misunderstanding lies in assuming that a specific field must be responsible for the existence of masses. You are forgetting that ins special relativity, masses arise from the "length" of the addition of two four vectors. Thus, the pi0 which has a mass of 135 MeV can decay in two photons of mass zero. The four vectors added of these photons have to be such as to add up to the invariant mass of the pi0.
And why isn't one field alone responsible for particles gaining mass?
The Higgs field is responsible for the mass of elementary particles of the standard model. All other measured masses of composite particles (protons, neutrons, pions, ...all the zoo of resonances, molecules etc) are the result of complicated four-vector additions which build up the invariant mass of the complex.
