Text on Left of Arrow in Flow Diagram

In order to make the text next to the downward pointing arrow appear left of the arrow instead of on the right, you can use swap:

From the pgf manual:

documentclass{article}
usepackage{tikz}
usetikzlibrary{chains, arrows.meta}
begin{document}

begin{center}

 begin{tikzpicture}[auto,
    node distance = 3mm and 2mm,
      start chain = A going below,
       arr/.style = {-Stealth},
every edge/.style = {draw, arr},
        BC/.style = {decorate,  % Brace Calligraphic
                     decoration={calligraphic brace, amplitude=2mm,
                                 raise=1mm, mirror},
                     very thick, pen colour={black}
                    },
       box/.style = {draw, rounded corners, fill=blue!20, align=center,
                     minimum height=5ex, text width=5.5em},
  decision/.style = {diamond, aspect=1.5, draw, fill=blue!20,
                    inner xsep=-3pt, minimum width=5.4em, align=center},
    ellip/.style = { ellipse, aspect=1.5,draw, fill=blue!20,
                    inner xsep=-3pt, text width=5.4em, align=center},
                        ]
% nodes in chain
    begin{scope}[nodes={on chain=A, join=by arr}]
node   [box]     {start};         % name=A-1
node   [box]       {end};

    end{scope}

 draw[arr, swap] (A-1.south) -- node{This Text} (A-2.north);

end{tikzpicture}
    end{center}


end{document}