Like yourself, I have found a reference for these results hard to find, even in the case of Hida families. These are a few pointers.

In the case of Hida families, once the Atkin-Lehner involutions are given an appropriate group-theoretic or geometric definition, the same proof that modular forms interpolate implies that the involutions interpolate. The geometric argument is given for instance in Mazur-Wiles (84) page 237 though of course the application to Hida theory is not worked at there for obvious historical reasons. The group-theoretic argument in the case $q=p$ (which you exclude, but which is strictly harder) is given in Nekovar-Platter (2000) section 1.6. It seems to me there might be everything you need in Ohta (Composition 99) section 3.4

That said, my experience has been that there are so many conflicting normalizations about everything in the literature that it is probably safer not to look at any reference and just reprove the results needed with one's own choices of normalizations (this is what I ended up doing in a slightly more general framework in my article at Compositio).

For Coleman families, the results of Bellaïche-Chenevier (2009) section 7.4 plus an appropriate definition of the Atkin-Lehner operators purely in terms of local automorphic representations seem a good place to start (presumably, Buzzard's formalization does the job equally well but I am less familiar with it). Just plug in in your test objects the supplementary requirement that there are operators satisfying the appropriate properties and the uniqueness of the eigenvariety will then imply that on sufficiently small affinoids, there exists similar operators on the sheaf of admissible $\mathbf{G}(\mathbb A_{\mathbb Q})$-representations. It seems also probable that that Mazur-Wiles calculations for the Igusa tower plus the construction of the eigencurve through the Igusa tower yields the results, so I guess that Pilloni (Ann. Institut Fourier 2013?) plus Mazur-Wiles might also be good starting points.

Please do provide your own answer if you ever find the perfect reference...