By default, the box that contains the part number has fixed width, but we can change that.
Using a dynamic box with prescribed minimum width, like the one provided in this answer, we can automatically move the part title to the right if the part number doesn't fit in the designated space.
The following code puts the part number inside a box whose size is either the default size (which aligns with other entries in the ToC) or the size needed to typeset the part number and a single space, whichever is larger.
\newcommand*{\minwidthbox}[2]{\makebox[{\ifdim#1<\width\width\else#1\fi}][l]{#2}}
\renewcommand*{\partnumberlinebox}[2]{\minwidthbox{#1}{#2\space}}
(\space is a basic LaTeX macro which expands to a single space. You may replace it with some other horizontal space as you see fit.)
The resulting ToC will look something like this (vertical line added in GIMP for clarity):

To reproduce this example:
\documentclass{memoir}
\chapterstyle{crosshead}
\newcommand{\minwidthbox}[2]{\makebox[{\ifdim#1<\width\width\else#1\fi}][l]{#2}}
\renewcommand{\partnumberlinebox}[2]{\minwidthbox{#1}{#2\space}}
\begin{document}
\vspace{-2cm}
\tableofcontents
\part{Narrow Roman numeral -- title aligned with chapters}
\chapter{Chapter}
\section{Section}
\chapter{Chapter}
\part{Narrow Roman numeral -- title aligned with chapters}
\chapter{Chapter}
\section{Section}
\chapter{Chapter}
\part{Wider Roman numeral -- title separated by space}
\chapter{Chapter}
\section{Section}
\chapter{Chapter}
\part{Wider Roman numeral -- title separated by space}
\chapter{Chapter}
\section{Section}
\chapter{Chapter}
\part{Narrow Roman numeral -- title aligned with chapters}
\chapter{Chapter}
\section{Section}
\chapter{Chapter}
\setcounter{part}{7}
\part{Very wide Roman numeral -- title separated by space}
\chapter{Chapter}
\section{Section}
\chapter{Chapter}
\end{document}