In case anyone else using the MNRAS LaTeX class has been tearing their hair out inserting line breaks manually into long author (and institution) lists, using \newauthor, here's how I did it:

\author[J.~Bloggs et al.]
{\parbox{\textwidth}{J.~Bloggs,$^{1}$\thanks{E-mail: \texttt{j.bloggs@no.where}}
J.~Bloggs,$^{2,3}$
J.~Bloggs,$^{3}$
J.~Bloggs,$^{4}$
J.~Bloggs,$^{5}$
J.~Bloggs,$^{6}$
%...
J.~Bloggs$^{99}$ and
J.~Bloggs$^{100}$}\vspace{0.4cm}\\
\parbox{\textwidth}{$^{1}$Department of Something, Somewhere\\
$^{2}$Department of Something, Somewhere\\
$^{3}$Department of Something, Somewhere\\
$^{4}$Department of Something, Somewhere\\
$^{5}$Department of Something, Somewhere\\
$^{6}$Department of Something, Somewhere\\
%...
$^{99}$Department of Something, Somewhere\\
$^{100}$Department of Something, Somewhere}}

Next: a Python script to generate the above from a simple list of authors and affiliations.