The answer to the question

• How to change the chapter number to display as text (One, Two, …)?

shows a way changing the chapter number in all places:

\usepackage{fmtcount}
\renewcommand{\thechapter}{\Numberstring{chapter}}

However, this affects also cross-references, and section numbering such as poducing One.1 instead of 1.1, which is consistent could be undesired. If you would like to change the prefix and number in the headings, not changing the general presentation of the counter, you could redefine \chapterformat. Furthermore, if you would like to change the marks in the header in the same way, you could adjust \chaptermarkformat.

\documentclass[chapterprefix=true]{scrreprt}
\usepackage{fmtcount}
\renewcommand*{\chapterformat}{%
  \mbox{\chapappifchapterprefix{\nobreakspace}\Numberstring{chapter}\autodot\enskip}%
}
\renewcommand*\chaptermarkformat{\chapappifchapterprefix{\ }%
  \Numberstring{chapter}\autodot\enskip}
\pagestyle{headings}
\begin{document}
\chapter{First Chapter}
\label{chap:1}
This is chapter \ref{chap:1}.
\clearpage
text
\clearpage
text
\end{document}​

The answer to the question

• How to change the chapter number to display as text (One, Two, …)?

shows a way changing the chapter number in all places:

\usepackage{fmtcount}
\renewcommand{\thechapter}{\Numberstring{chapter}}

However, this affects also cross-references, and section numbering such as poducing One.1 instead of 1.1, which is consistent could be undesired. If you would like to change the prefix and number in the headings, not changing the general presentation of the counter, you could redefine \chapterformat. Furthermore, if you would like to change the marks in the header in the same way, you could adjust \chaptermarkformat.

\documentclass[chapterprefix=true]{scrreprt}
\usepackage{fmtcount}
\renewcommand*{\chapterformat}{%
  \mbox{\chapappifchapterprefix{\nobreakspace}\Numberstring{chapter}\autodot\enskip}%
}
\renewcommand*\chaptermarkformat{\chapappifchapterprefix{\ }%
  \Numberstring{chapter}\autodot\enskip}
\pagestyle{headings}
\begin{document}
\chapter{First Chapter}
\label{chap:1}
This is chapter \ref{chap:1}.
\clearpage
text
\clearpage
text
\end{document}​