How to manually adjust symbol positioning in `\underset`?

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Edit # 2

To raise the whole operator, could use \scalebox from the graphics package. This translates better, as is found independent of the font size option of the documentclass (the original version doesn't scale equally well to a 10pt or 11pt document)

\documentclass{standalone}
\usepackage{amsmath, graphics}
\newcommand{\upperset}[2]{\:
  \underset{
            \text{\raisebox{1.2ex}{\smash{\scalebox{0.8}{$#1$}}}}%
           }
           {
            \text{\raisebox{0.2ex}{\smash{$#2$}}}
           }
                       \:}
\begin{document}
    $x \underset{{}^u}{\sim} y$ \quad\phantom{\strut}
    $x \upperset{{}^u}{\sim} y$
\end{document}

giving very near to what you wanted:

Original post

Maybe this could be a possible solution:

\documentclass[12pt]{standalone}
\usepackage{amsmath}
\newcommand{\upperset}[2]{%
    \underset{%
        \text{\raisebox{1ex}{\smash{\fontsize{5}{5}$#1$}}}
              }{#2}%
                         }
\begin{document}
    $x \underset{{}^u}{\sim} y$ \quad\phantom{\strut}
    $x \upperset{{}^u}{\sim} y$
\end{document}

which gives this:

If I understand correctly, the OP wants the underset to sit on the baseline.

\documentclass{article}
\usepackage{stackengine,amsmath}
\newcommand\runderset[2][\sim]{\mathrel{\ensurestackMath{%
  \stackengine{-.2pt}{\scriptscriptstyle#2}{#1}{O}{c}{F}{F}{S}}}}
\begin{document}
$a\runderset{x}b$

$a\runderset[=]y b$
\end{document}