Here's a 2.0 option:
EDIT: Sorry...the 1.0 requirement was added after I started on my answer.
XML
<?xml version="1.0" encoding="UTF-8"?>
<doc>
<Name>O'Niel</Name>
<Name>St Peter</Name>
<Name>A.David</Name>
</doc>
XSLT 2.0
<?xml version="1.0" encoding="UTF-8"?>
<xsl:stylesheet version="2.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform">
<xsl:output indent="yes"/>
<xsl:strip-space elements="*"/>
<xsl:template match="*|@*">
<xsl:copy>
<xsl:apply-templates select="node()|@*"/>
</xsl:copy>
</xsl:template>
<xsl:template match="text()">
<xsl:value-of select="replace(.,'[^a-zA-Z]','')"/>
</xsl:template>
</xsl:stylesheet>
Output
<?xml version="1.0" encoding="UTF-8"?>
<doc>
<Name>ONiel</Name>
<Name>StPeter</Name>
<Name>ADavid</Name>
</doc>
Here are a couple more ways of using replace()
...
Using "i" (case-insensitive mode) flag:
replace(.,'[^A-Z]','','i')
Using category escapes:
replace(.,'\P{L}','')