You can store the original definition of the several dots commands before loading unicode-math and restore them inside \begin{document}:

\documentclass{article}
\NewCommandCopy{\latexvdots}{\vdots}
\NewCommandCopy{\latexddots}{\ddots}
\NewCommandCopy{\latexcdots}{\cdots}
\usepackage{unicode-math}
\AtBeginDocument{%
  %\ShowCommand\mathellipsis
  \renewcommand\mathellipsis{\mathinner{\ldotp\ldotp\ldotp}}%
  %\ShowCommand\mathellipsis
  %\ShowCommand\vdots\ShowCommand\latexvdots
  \RenewCommandCopy{\vdots}{\latexvdots}%
  %\ShowCommand\ddots\ShowCommand\latexddots
  \RenewCommandCopy{\ddots}{\latexddots}%
  %\ShowCommand\cdots\ShowCommand\latexcdots
  \RenewCommandCopy{\cdots}{\latexcdots}%
}
\begin{document}
$\dots, \ldots, \cdots, \vdots, \ddots$
\end{document}

If you want to see the different definitions, you can activate the \ShowCommand lines. They will show the definitions in the log file and on the terminal.

You can store the original definition of the several dots commands before loading unicode-math and restore them inside \begin{document}:

\documentclass{article}
\NewCommandCopy{\latexvdots}{\vdots}
\NewCommandCopy{\latexddots}{\ddots}
\NewCommandCopy{\latexcdots}{\cdots}
\NewCommandCopy{\latexmathellipsis}{\mathellipsis}
\usepackage{unicode-math}
\AtBeginDocument{%
  %\ShowCommand\mathellipsis\ShowCommand\latexmathellipsis
  \RenewCommandCopy{\mathellipsis}{\latexmathellipsis}%
  %\ShowCommand\vdots\ShowCommand\latexvdots
  \RenewCommandCopy{\vdots}{\latexvdots}%
  %\ShowCommand\ddots\ShowCommand\latexddots
  \RenewCommandCopy{\ddots}{\latexddots}%
  %\ShowCommand\cdots\ShowCommand\latexcdots
  \RenewCommandCopy{\cdots}{\latexcdots}%
}
\begin{document}
$\dots, \ldots, \cdots, \vdots, \ddots$
\end{document}

If you want to see the different definitions, you can activate the \ShowCommand lines. They will show the definitions in the log file and on the terminal.

You can store the original definition of the several dots commands before loading unicode-math and restore them inside \begin{document}:

\documentclass{article}
\NewCommandCopy{\latexvdots}{\vdots}
\NewCommandCopy{\latexddots}{\ddots}
\NewCommandCopy{\latexcdots}{\cdots}
\usepackage{unicode-math}
\AtBeginDocument{%
  \renewcommand\mathellipsis{\mathinner{\ldotp\ldotp\ldotp}}%
  \RenewCommandCopy{\vdots}{\latexvdots}%
  \RenewCommandCopy{\ddots}{\latexddots}%
  \RenewCommandCopy{\cdots}{\latexcdots}%
}
\begin{document}
$\dots, \ldots, \cdots, \vdots, \ddots$
\end{document}

You can store the original definition of the several dots commands before loading unicode-math and restore them inside \begin{document}:

\documentclass{article}
\NewCommandCopy{\latexvdots}{\vdots}
\NewCommandCopy{\latexddots}{\ddots}
\NewCommandCopy{\latexcdots}{\cdots}
\usepackage{unicode-math}
\AtBeginDocument{%
  %\ShowCommand\mathellipsis
  \renewcommand\mathellipsis{\mathinner{\ldotp\ldotp\ldotp}}%
  %\ShowCommand\mathellipsis
  %\ShowCommand\vdots\ShowCommand\latexvdots
  \RenewCommandCopy{\vdots}{\latexvdots}%
  %\ShowCommand\ddots\ShowCommand\latexddots
  \RenewCommandCopy{\ddots}{\latexddots}%
  %\ShowCommand\cdots\ShowCommand\latexcdots
  \RenewCommandCopy{\cdots}{\latexcdots}%
}
\begin{document}
$\dots, \ldots, \cdots, \vdots, \ddots$
\end{document}

If you want to see the different definitions, you can activate the \ShowCommand lines. They will show the definitions in the log file and on the terminal.