Parsing strings containing diacritical marks (macros?)

The following works with the input described above.

\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn
\NewDocumentCommand{\getargs}{m}{
    \get_args:n {#1}
}
\int_new:N \narg
\cs_new_protected:Npn \get_args:n #1 {
    \seq_set_split:Nnn \l_tmpa_seq {~} {#1}
        \seq_map_inline:Nn \l_tmpa_seq {
            \int_incr:N \l_tmpa_int
            \tl_set:cx {arg\int_to_roman:n {\l_tmpa_int}}{##1}
        }
    \int_set:Nn \narg {\l_tmpa_int}
    \int_zero:N \l_tmpa_int
    }
\ExplSyntaxOff
\begin{document}
\getargs{mein hut} \argi~\argii $<$NO PROBLEM\\
BUT PUT UMLAUT ON THE u AND IT BREAKS
\getargs{mein h\"ut} \argi~\argii
\end{document}

Here is a solution with no expansion, able to parse macros and groups {...}:

\documentclass{article}
\makeatletter
\newcount\arg@index
\newtoks\code@toks

\def\getargsF#1{%
    \arg@index=1
    \code@toks{}%
    \parse@i#1\parse@stop
}

\def\add@tok#1{\code@toks\expandafter{\the\code@toks#1}}

\def\parse@stop{\parse@stop}

\def\parse@i{\futurelet\nxttok\parse@ii}

\def\parse@ii{%
    \ifx\nxttok\parse@stop \let\next@action\parse@stop@i
    \else
        \ifx\nxttok\@sptoken \let\next@action\read@space
        \else
            \ifx\nxttok\bgroup \let\next@action\read@bracearg
            \else \let\next@action\testtoken
            \fi
        \fi
    \fi
    \next@action
}

\def\parse@stop@i\parse@stop{\assign@arg}

\expandafter\def\expandafter\read@space\space{%
    \assign@arg
    \advance\arg@index1
    \code@toks{}%
    \parse@i
}

\def\read@bracearg#1{%
    \add@tok{{#1}}%
    \parse@i
}

\def\testtoken#1{%
    \if\noexpand~\noexpand#1%
        \ifnum\catcode`#1=\active
            \assign@arg
            \advance\arg@index1
            \code@toks{}%
        \else
            \add@tok{#1}%
        \fi
    \else
        \add@tok{#1}%
    \fi
    \parse@i
}

\def\assign@arg{%
    \expandafter\edef\csname arg\romannumeral\arg@index\endcsname{\the\code@toks}%
}
\makeatother
\begin{document}
\getargsF{abcd efgh~h\"ut}\argi\#\argii\#\argiii\#

\getargsF{maths $1+1=2$ \textbf{b\textit{ol}d} \^a\'i\'o~hardspc~end}\argi\#\argii\#\argiii\#\argiv\#\argv\#\argvi\#

\catcode`\~12
\getargsF{maths $1+1=2$ \textbf{b\textit{ol}d} \^a\'i\'o~hardspc~end}\argi\#\argii\#\argiii\#\argiv\#
\end{document}

which gives

Once you have expanded everything in the original input with \protected@edef you really don't need to step through token by token which makes it hard to handle any kind of brace group or command taking arguments. This just scans with a delimited argument looking for spaces which is much simpler.

 \documentclass{article}
\makeatletter

\def\string@end{$\SaveHardspace}
\def\converttilde{T}
\newcounter{arg@index}
\let\SaveHardspace~%%%
\def\getargsF#1{%
  \if T\converttilde\def~{ }\fi
  \protected@edef\the@string{#1}%
  \setcounter{arg@index}{0}%
  \lowercase{\expandafter\parse@Block\the@string} \string@end
  \let~\SaveHardspace
}

\def\parse@Block#1 {%
  \stepcounter{arg@index}%
  \@namedef{arg\roman{arg@index}}{#1}%
  \futurelet\tmp\parse@Block@}

\def\parse@Block@{%
\ifx\tmp\string@end\edef\narg{\thearg@index}\expandafter\@gobble
\else\expandafter\parse@Block\fi}

\makeatother
\begin{document}
\getargsF{mein hut} \argi~\argii~[\narg] $<$ NO PROBLEM


BUT PUT UMLAUT ON THE u AND IT BREAKS

\getargsF{mein h\"ut see}  \argi~\argii~[\narg]
\end{document}