The template does not work together with biblatex, in fact it loads a modified (!) version of natbib. So I strongly advise against trying to get biblatex to work and use natbib instead. If you are submitting to that journal it is quite unlikely anyway that they can accept biblatex submissions, see Biblatex: submitting to a journal
The following MWE worked fine for me on Overleaf, see https://www.overleaf.com/read/nvzmmrbkmtvd
\documentclass[a4,center,fleqn]{NAR}
\copyrightyear{2008}
\pubdate{31 July 2009}
\pubyear{2009}
\jvolume{37}
\jissue{12}
\begin{document}
\title{Article title}
\author{Someone}
\address{Location}
\history{None}
\maketitle
\begin{abstract}
Text.
\end{abstract}
\section{Text}
\cite{wilde,kullback,moore}
\bibliographystyle{plainnat}
\bibliography{biblatex-examples}
\end{document}
That is not to say that there is absolutely no way to get biblatex to work with that template (there is), but you should not go down that route. The particular problem is that the class defines a few commands that biblatex needs to define itself and makes the bold move
\mathchardef\@m=1500 % adapted value
which seems like a really bad idea to me, since biblatex (and other packages as well) assumes the standard definition of
\mathchardef\@m=1000
This needs to be fixed so that you end up loading the class with the option openbib to avoid natbib and
\usepackage{etoolbox}
\undef\bibfont
\makeatletter
\mathchardef\@m=1000
\makeatother
\usepackage{biblatex}
to fix its definitions.