pgfplotstable: How to access column name?

As @percusse stated correctly, the root challenge here is that these columns/<name>/.style things are only interpreted in the context of \pgfplotstabletypeset.

A solution which is also a direct answer to your question and solves the MWE means to query the information from the correct context. This is simpler than it sounds and I can give such an example below. It does not need ANY knowledge of internals of pgfplotstable. It only relies on pgfkeys and basic TeX expansion control.

Here is the solution:

\documentclass{article}
\usepackage{amsmath}
\usepackage{pgfplots, pgfplotstable, filecontents}
\pgfplotsset{compat=1.6}

\begin{filecontents}{test.dat}
  Time Distance
   0 0 
   1 1 
   3 2 
\end{filecontents}

\pgfplotstableset{
   columns/Time/.style={
     column name={$t_{\alpha}$},
   },
   columns/Distance/.style={
      column name={$D_{\alpha}$},
   },
}

% defines \devendraretval to contain the display name for the column
% named '#1' 
%
% #1: a column name
\def\getcolumndisplayname#1{%
     % we WANT a group. Otherwise, any values set in these style would
     % become global and they would be mixed up between different
     % columns.
     \begingroup
         % Access the contents of column's display name programmatrically:
        \pgfplotstableset{columns/#1/.try}%
        \pgfkeysgetvalue{/pgfplots/table/column name}\temp
        \ifx\temp\empty
            % oh. We have no such value! Ok, use the column name as
            % such.
            \edef\temp{#1}%
        \fi
        % this here is one of the possible patterns to communicate a
        % local variable outside of a local group:
        \global\let\GLOBALTEMP=\temp
     \endgroup
     \let\devendraretval=\GLOBALTEMP
     %
}%

\begin{document}
\thispagestyle{empty}
  \pgfplotstableread{test.dat}\loadedtable

  \pgfplotstabletypeset{\loadedtable}

  \pgfplotstableforeachcolumn\loadedtable\as\col{%
     %
     \getcolumndisplayname{\col}%
     column name is ‘\col’; index is \pgfplotstablecol; display name is `\devendraretval'\par
  }

  \getcolumndisplayname{Time}%
  Value of Time is \devendraretval.
\end{document}

It defines a new command `\getcolumndisplayname{}' which retrieves it.

Please refer to http://pgfplots.sourceforge.net/TeX-programming-notes.pdf for details about the basic instructions.

Alternative from my comment (and from Use first row of a table as legend entry in pgfplot graph?):

\documentclass[border=1pt]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.6}

\pgfplotstableread{
time distance velocity something
0 0 1 0.2
1 1 1 0.3
1.999 1.999 1 0.4
2 2 0 0.4
3 2 0 0.5
}\mytable

\pgfplotstableread{
name alias
time $t$
distance $d_\alpha$
velocity $\sin(2t)$
something nothing
}\tablenames

\pgfplotstablegetcolsof{\mytable}
\pgfmathtruncatemacro\numberofcols{\pgfplotsretval-1}

\begin{document}
\begin{tikzpicture}
\begin{axis}
\pgfplotsinvokeforeach{1,...,\numberofcols}{
  \addplot table [y index=#1] \mytable;
  \pgfplotstablegetelem{#1}{[index] 1}\of{\tablenames}
  \addlegendentryexpanded{\pgfplotsretval} 
}
\end{axis}
\end{tikzpicture}
\end{document}