XSLT Transform doesn't work until I remove root node

The problem: your XML puts its elements in a namespace.

Solution: declare the same namespace in your stylesheet, assign it a prefix and use that prefix to address the elements in the source XML:

XSLT 1.0

<xsl:stylesheet version="1.0"
xmlns:xsl="http://www.w3.org/1999/XSL/Transform"
xmlns:met="www.metoffice.gov.uk/xml/metoRegionalFcst"
exclude-result-prefixes="met">
<xsl:template match="/">
  <html>
  <body>
   <xsl:value-of select="met:RegionalFcst/met:FcstPeriods/met:Period/met:Paragraph[@title='Headline:']"/>
  </body>
  </html>
</xsl:template>
</xsl:stylesheet>