The standard method for investigating decay of correlations of expanding maps is using operator theory, either directly or via an induced dynamical system. The decay of correlations can then be deduced from the spectral properties of the Ruelle transfer operator acting on the space of Hoelder functions, possibly in combination with operator renewal theory if the expansion is not uniform. However, in your case this would require applying an arbitrary sequence of distinct transfer operators, and knowing the spectral properties of these operators would not be of immediate help.
However, since you are interested specifically in the linear expanding maps $x \mapsto nx \mod 1$, I suggest that you argue directly using Fourier analysis. If you restrict $v$ to a high enough smoothness category ($C^k$ for some sufficiently large $k$, or perhaps even real analytic) then you should be able to obtain a correlation estimate in a fairly direct manner by exploiting the rate of decay of the Fourier coefficients of $v$.