We study the effect of a quantum quench between two tunnel coupled Tomonaga-Luttinger liquids (TLLs) with different speed of sound and interaction parameter. The quench dynamics is induced by switching off the tunneling and letting the two systems evolve independently. We fully diagonalize the problem within a quadratic approximation for the initial tunneling. Both the case of zero and finite temperature in the initial state are considered. We focus on correlation functions associated with the antisymmetric and symmetric combinations of the two TLLs (relevant for interference measurements), which turn out to be coupled due to the asymmetry in the two systems' Hamiltonians. The presence of different speeds of sound leads to multiple light cones separating different decaying regimes. In particular, in the large time limit, we are able to identify a prethermal regime where the two-point correlation functions of vertex operators of symmetric and antisymmetric sector can be characterized by two emerging effective temperatures, eventually drifting towards a final stationary regime that we dubbed quasithermal, well approximated at large scale by a thermal-like state, where these correlators become time independent and are characterized by a unique correlation length. If the initial state is at equilibrium at nonzero temperature T0, all the effective temperatures acquire a linear correction in T0, leading to faster decay of the correlation functions. Such effects can play a crucial role for the correct description of currently running cold atoms experiments.