How can I use \SI from siunitx inside \replaced from changes?

The problem here is actually an interaction with the \sout command from the ulem package. Essentially, there is a clash over the insertion of a \penalty between the number and the unit: ulem alters how this expects to work and that doesn't agree with the box/grouping in siunitx. I've not traced it all back to a minimal example of the low-level code, but that probably wouldn't help in any case (at least without quite a bit of effort). What you can do is simply disable the insertion of a penalty. There isn't an interface for this (one of the few things not in the options for siunitx!) so at the moment a low-level redefinition is required

\documentclass{article}
\usepackage{siunitx}
\usepackage{changes}
\ExplSyntaxOn
\cs_set_eq:NN \__siunitx_unit_output_number_sep:
  \__siunitx_unit_output_number_sep_aux:
\ExplSyntaxOff
\begin{document}
    \replaced{\SI{1}{\volt}}{\SI{2}{\volt}}
\end{document}