As a complement to Peter's answer, let me try to address the precise question of whether the periodic cyclic homologies of the almost categories are equivalent. The answer is "no". First, what is the definition of the periodic cyclic homology of the almost category? This is not obvious, but Efimov has shown the way (though perhaps his work post-dates your question?). Let me start by recalling his theory, which he calls "continuous K-theory". For more details of what I'm writing you can see Marc Hoyois' nice exposition at http://www.mathematik.ur.de/hoyois/papers/efimov.pdf .

First, any invariant of (idempotent-complete $\mathbb{Z}$-linear, but let me not carry these adjectives around) small stable $\infty$-categories, for example periodic cyclic homology, can equivalently be viewed as an invariant of compactly generated stable $\infty$-categories, by taking compact objects. If we want to take care of functoriality we should posit that we only consider those functors between compactly generated $\infty$-categories which preserve the compact objects; equivalently, and this will be better for what's coming next, those functors whose right adjoints also preserve colimits.

Now, there is a substantial broadening of the notion of compactly generated table $\infty$-category due to Lurie, the notion of a dualizable stable $\infty$-category. Every dualizable stable $\infty$-category $\mathcal{C}$ is the kernel of a localization functor between compactly generated stable $\infty$-categories, where again the right adjoint to the localization functor is required to preserve all colimits; and actually this is a precise characterization of the dualizable stable $\infty$-categories.

This leads to Efimov's fantastic result (previsaged to some extent by Tamme's work on excision, see https://arxiv.org/abs/1703.03331): if $E$ is a localizing invariant, then there is a unique functorial extension of $E$ to a localizing invariant of dualizable stable $\infty$-categories. This tells you what $E(\mathcal{C})$ has to be, if $\mathcal{C}$ is presented as the kernel of a localization functor between compactly generated guys as above: just the fiber of $E$ applied to that localization functor.

The almost category provides a good example, because it more or less explicitly arises as the kernel of such a localization functor, namely $\operatorname{Mod}(\mathcal{O}_C)^a$ is the kernel of the base-change functor $\operatorname{Mod}(\mathcal{O}_C)\rightarrow \operatorname{Mod}(k)$ where $k$ is the residue field. The weird property of the residue field that makes this base-change functor a localization is that the derived tensor product of $k$ with itself over $\mathcal{O}_C$ is $k$ again. This is what makes almost mathematics run.

Summing up, what this shows is that the perodic cyclic homology of the almost category is basically the same as the periodic cyclic homology of the usual category of modules. They only differ by the periodic cyclic homology of the residue field, which is anyway the same for $\mathcal{O}_C$ and its tilt.

You also ask if something weaker could be true, that these invariants of $R$ and its tilt are "related by a spectral sequence". I guess there's a bit a freedom in interpreting what this means, but I suppose with appropriate definitions and interpretations the answer should be "yes". More precisely, if $\pi$ and $\pi^\flat$ are as in Peter's answer, then the associated gradeds for the $\pi$-adic and $\pi^\flat$ adic filtrations are isomorphic (provided, let's say, that $\pi$ and $\pi^\flat$ are non-zerodivisors). If we agree to replace periodic cyclic homology with its "continuous" variant (now not in the sense of Efimov but in the sense of the inverse limit of the periodic cyclic homologies of the quotients by the stages in the $\pi$-adic filtration... though this is also conjecturally a special case of the Efimov construction, applied to the so-called nuclear modules which Peter and I have defined), then it's easy to imagine that there should be a spectral sequence converging from the invariant for the associated graded to the invariant for the ring, so that indeed the two things may well be related by two spectral sequences with the same $E_2$-page.