Thanks to everyone above, but this is now fixed. For the benefit of others the most significant error was in aligning the three namespaces as suggested by Ian.
For completeness, here is the corrected XML and XSD
Here is the XML, with the typos corrected (sorry for any confusion caused by tardiness)
<?xml version="1.0" encoding="UTF-8"?>
<Root xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xmlns="urn:Test.Namespace"
xsi:schemaLocation="urn:Test.Namespace Test1.xsd">
<element1 id="001">
<element2 id="001.1">
<element3 id="001.1" />
</element2>
</element1>
</Root>
and, here is the Schema
<?xml version="1.0"?>
<xsd:schema xmlns:xsd="http://www.w3.org/2001/XMLSchema"
targetNamespace="urn:Test.Namespace"
xmlns="urn:Test.Namespace"
elementFormDefault="qualified">
<xsd:element name="Root">
<xsd:complexType>
<xsd:sequence>
<xsd:element name="element1" maxOccurs="unbounded" type="element1Type"/>
</xsd:sequence>
</xsd:complexType>
</xsd:element>
<xsd:complexType name="element1Type">
<xsd:sequence>
<xsd:element name="element2" maxOccurs="unbounded" type="element2Type"/>
</xsd:sequence>
<xsd:attribute name="id" type="xsd:string"/>
</xsd:complexType>
<xsd:complexType name="element2Type">
<xsd:sequence>
<xsd:element name="element3" type="element3Type"/>
</xsd:sequence>
<xsd:attribute name="id" type="xsd:string"/>
</xsd:complexType>
<xsd:complexType name="element3Type">
<xsd:attribute name="id" type="xsd:string"/>
</xsd:complexType>
</xsd:schema>
Thanks again to everyone, I hope this is of use to somebody else in the future.