  We study theoretically the \emph{return probability experiment}, used to
measure the dephasing time $T_2^*$, in a double quantum dot (DQD) in
semiconducting carbon nanotubes (CNTs) with spin-orbit coupling and disorder
induced valley mixing. Dephasing is due to hyperfine interaction with the spins
of the ${}^{13}$C nuclei. Due to the valley and spin degrees of freedom four
bounded states exist for any given longitudinal mode in the quantum dot. At
zero magnetic field the spin-orbit coupling and the valley mixing split those
four states into two Kramers doublets. The valley mixing term for a given dot
is determined by the intra-dot disorder and therefore the states in the Kramers
doublets belonging to different dots are different. We show how nonzero
single-particle interdot tunneling amplitudes between states belonging to
different doublets give rise to new avoided crossings, as a function of
detuning, in the relevant two particle spectrum, crossing over from the two
electrons in one dot states configuration, $(0,2)$, to the one electron in each
dot configuration, $(1,1)$. In contrast to the clean system, multiple
Landau-Zener processes affect the separation and the joining stages of each
single-shot measurement and they affect the outcome of the measurement in a way
that strongly depends on the initial state. We find that a well-defined return
probability experiment is realized when, at each single-shot cycle, the (0,2)
ground state is prepared. In this case, valley mixing increases the saturation
value of the measured return probability, whereas the probability to return to
the (0,2) ground state remains unchanged. Finally, we study the effect of the
valley mixing in the high magnetic field limit; for a parallel magnetic field
the predictions coincide with a clean nanotube, while the disorder effect is
always relevant with a magnetic field perpendicular to the nanotube axis.
