You can use \getnamereftext and, of course, nameref (automatically loaded by hyperref).
\documentclass{book}
\usepackage{amsthm}
\usepackage{nameref}% or hyperref
\newtheorem{theorem}{Theorem}[chapter]
\newtheorem{example}[theorem]{Example}
\newtheorem*{continuancex}{Continuance of Example \continuanceref}
\makeatletter
\newcommand{\getnamereftext}[1]{%
@ifundefined{r@#1}{}{%
\unexpanded\expandafter\expandafter\expandafter{%
\expandafter\expandafter\expandafter@thirdoffive\csname r@#1\endcsname
}%
}%
}
\makeatletter
\newenvironment{continuance}[1]{%
\newcommand\continuanceref{\ref{#1}}%
\if\relax\getnamereftext{#1}\relax
\continuancex
\else
\continuancex[\nameref{#1}]%
\fi
}{\endcontinuancex}
\begin{document}
\chapter{Title}
\begin{example}\label{ex0}
This example has no title
\end{example}
Some text in between
\begin{continuance}{ex0}
Something else
\end{continuance}
\begin{example}[Description] \label{ex1}
This is an example.
\end{example}
Some text stands here. And then there is some space between this and the continued example.
Further should the continued example not be intended but start at the beginning of the line.
\begin{continuance}{ex1}
This is the continued example.
\end{continuance}
Some more text, which should follow after a little vertical space.
\end{document}

With no \expandafter and friends:
\documentclass{book}
\usepackage{amsthm}
\usepackage{nameref}% or hyperref
\newtheorem{theorem}{Theorem}[chapter]
\newtheorem{example}[theorem]{Example}
\newtheorem*{continuancex}{Continuance of Example \continuanceref}
\newcommand{\continuanceref}{}
\ExplSyntaxOn
\NewExpandableDocumentCommand{\getnamereftext}{m}
{
\tl_if_exist:cT { r@#1 }% check whether \ref is known
{% in this case extract the third item
\tl_item:cn { r@#1 } { 3 }
}
}
\NewDocumentEnvironment{continuance}{m}
{
\renewcommand\continuanceref{\ref{#1}}%
\tl_if_blank:eTF { \getnamereftext{#1} }
{ \continuancex }
{ \continuancex[\nameref{#1}] }
}
{\endcontinuancex}
\ExplSyntaxOff
\begin{document}
\chapter{Title}
\begin{example}\label{ex0}
This example has no title
\end{example}
Some text in between
\begin{continuance}{ex0}
Something else
\end{continuance}
\begin{example}[Description] \label{ex1}
This is an example.
\end{example}
Some text stands here. And then there is some space between this and the continued example.
Further should the continued example not be intended but start at the beginning of the line.
\begin{continuance}{ex1}
This is the continued example.
\end{continuance}
Some more text, which should follow after a little vertical space.
\end{document}
\begin{example}[title]...\end{example}. – Werner May 02 '23 at 16:20\newcommand{\exmaneA}{<repeatable text>}. – barbara beeton May 02 '23 at 16:27