Here's a solution that uses natbib and a modified version of apalike.bst. First, make a copy of apalike.bst and put it in the same folder as your thesis file. In this example, I'll name the file apalike-bilingual.
In it you will find two functions:
FUNCTION {format.lab.names}
and
FUNCTION {format.names}
In each of these functions you will find code that contains " and ". This is the code you will change. In the format.lab.names change:
{ " and " * s #2 "{vv~}{ll}" format.name$ * }
to
{ " \harvardand\ " * s #2 "{vv~}{ll}" format.name$ * }
In principle, this is the only thing you need to do, since you just want citation callouts to change. But for completeness, you also want to change the label-names function. Change:
{ " and " * t * }
to
{ " \harvardand\ " * t * }
What this does now makes a version of apalike that uses the \harvardand macro for its and. Note that the following \ in the changed code is important, since spaces get gobbled after a macro.
Using the solution that Ulrike Fischer provided here: language dependent citations and natbib we can now make the \harvardand command sensitive to the current babellanguage by adding the French version of \harvardand to \frenchcaptions. Thanks to Ulrike for this solution and to moewe for pointing out that we can use babel directly instead of using a conditional as in the linked question.
\begin{filecontents}{\jobname.bib}
@article{twoauthors,
title={A cool paper},
author={Duck, A. and Marmot, A.},
journal={Journal of Irrelevant Science},
year={2019},
volume={1}}
\end{filecontents}
\documentclass{article}
\usepackage{natbib}
\bibliographystyle{apalike-bilingual}
\usepackage[french,english]{babel}
\usepackage{etoolbox}
\DeclareRobustCommand{\harvardand}{and}
\appto\captionsfrench{\DeclareRobustCommand{\harvardand}{et}}
\begin{document}
English citation: \cite{twoauthors}
\begin{otherlanguage}{french}
French citation: \cite{twoauthors}
\end{otherlanguage}
\bibliography{\jobname}
\end{document}

biblatexwill of course change other aspects of the style as well, so there is still a risk of this not being exactly what was desired. – moewe Mar 31 '19 at 15:31