Do measurements of time-scales for decoherence disprove some versions of Copenhagen or MWI?

I am not aware of any experimental evidence, so this probably does not qualify as an answer. However I can offer a reference that addresses this question theoretically:

• Armen E. Allahverdyan, Roger Balian, Theo M. Nieuwenhuizen (2011) Understanding quantum measurement from the solution of dynamical models, https://arxiv.org/abs/1107.2138

and by the same group, but more recently:

• A.E. Allahverdyan, R. Balian, T.M. Nieuwenhuizen. (2017) A sub-ensemble theory of ideal quantum measurement processes. Annals of Physics, 376C, Sciencedaily URL, full article: https://arxiv.org/abs/1303.7257

Essentially they do what the OP describes in the question. They take a dynamical model of a macroscopic system and solve its unitary evolution within the Schrödinger equation. Then they try to look if some "measurement-like structure" emerges just from the many-body dynamics, without collapse.

There is one main difference to decorence, where usually only a system and an environment is considered (e.g. the Leggett-Caldeira model, also cf. wiki article on quantum dissipation). In the work mentioned above, a macroscopic system that mimics a detector is included. Like the environment this is also a macroscopic system, but unlike the environment it has some special properties that allow it to record information. In the first paper this is done by considering a ferro-magnet, whose spontaneous symmetry breaking allows it to have a macroscopic polarization, which is essentially a deterministic property after equilibration (simply because the flip probability is very low).

As far as I am aware this is far from a solution to the measurement problem, some open issues are mentioned in the articles themselves. At least it goes into the right direction however, especially it starts addressing the question of measurement timescales, which can maybe also pave the way for experimental investigations thereof.

Do measurements of time-scales for decoherence disprove some versions of Copenhagen or MWI?

No.

From Decoherence on wikipedia (emphasis mine):

Decoherence has been used to understand the collapse of the wavefunction in quantum mechanics. Decoherence does not generate actual wave function collapse. It only provides an explanation for the observation of wave function collapse, as the quantum nature of the system "leaks" into the environment. That is, components of the wavefunction are decoupled from a coherent system, and acquire phases from their immediate surroundings. A total superposition of the global or universal wavefunction still exists (and remains coherent at the global level), but its ultimate fate remains an interpretational issue. Specifically, decoherence does not attempt to explain the measurement problem. Rather, decoherence provides an explanation for the transition of the system to a mixture of states that seem to correspond to those states observers perceive.

As Wolpertinger said, to disprove Copenhagen or MWI you should challenge the postulate that the measurement act is instantaneous, by taking into account both detector and probe. I'm not an expert on this, so I cannot add much. I just wanted to point out that decoherence is not enough to solve the measurement problem.

Some further relevant quotes:

The discontinuous "wave function collapse" postulated in the Copenhagen interpretation to enable the theory to be related to the results of laboratory measurements cannot be understood as an aspect of the normal dynamics of quantum mechanics via the decoherence process. Decoherence is an important part of some modern refinements of the Copenhagen interpretation. Decoherence shows how a macroscopic system interacting with a lot of microscopic systems (e.g. collisions with air molecules or photons) moves from being in a pure quantum state—which in general will be a coherent superposition (see Schrödinger's cat)—to being in an incoherent improper mixture of these states. [...] However, decoherence by itself may not give a complete solution of the measurement problem, since all components of the wave function still exist in a global superposition, which is explicitly acknowledged in the many-worlds interpretation. All decoherence explains, in this view, is why these coherences are no longer available for inspection by local observers. To present a solution to the measurement problem in most interpretations of quantum mechanics, decoherence must be supplied with some nontrivial interpretational considerations [...]