How to remember a length across compilations

\def\reloadmylen#1{\global\mylen=#1}

\protected@write is used if one wants to write the current value of a macro at the time of page-shipout to an external file.

In the OP's case, the dimen value after use of \changelen shall be stored in \@auxout (\jobname.aux). This can be done immediately and doesn't need to be delayed further, since the last value written out by \changelen is what counts when \jobname.aux is re-read during the second LaTeX run. That is why \protected@write is not the best choice here, but \immediate\write should be preferred.

---

\documentclass{article}

\newlength\mylen
\mylen=0pt% Default value

\makeatletter    
% persistently sets length register
\def\changelen#1#2{%
  \global#1=#2%
  \immediate\write\@auxout{\global#1=\the#1}%
}
\makeatother

\begin{document}

\the\mylen% Prints 20pt after the second run.

\changelen{\mylen}{10pt}
%\changelen{\mylen}{20pt}
\newlength\otherlen\otherlen=20pt
\changelen{\mylen}{\otherlen} % also works (can be used like `\setlength')

\end{document}

Depending on the precision you need, you can simply use the totcount package:

\documentclass{article}

\usepackage{printlen}

\usepackage{totcount}
\newtotcounter{mylenght}


\begin{document}

\uselengthunit{mm}\printlength{\totvalue{mylenght}}

\newlength\mylen
\setlength{\mylen}{4.9999999mm}

\setcounter{mylenght}{\mylen}

\end{document}