What does it mean for a Wavelet transform to commute with translations?
We can start from what is "shift invarient":

Transform G is shift invariant if - $$\forall x:\sigma^nG(x) = G(x)$$
$\sigma^n$ being shift by n. Examples for transforms that are invarient to shifts are histogram and the amplitude of Fourier transform.

Commuting with shift is - $$\forall x:\sigma^nG(x) = G(\sigma^nx)$$
So it can't be shift invariant (unless G(x) is constant).

Note: I'm actually not certain that wavelet transform commutes with shift, but it's most certainly is not shift inveriant.