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.