Based on that code, you would refer to it as "URL.UDF.getMessage()".
Where does the line "cfset structAppend(url,createObject( "component", "UDF" ))" actually occur inside Application.cfc (above any function definitions or inside one of the functions)? If it's at the top of the component definition, that UDF gets instantiated on each request and put in that page's URL scope. Does the UDF component maintain state in any way? If not, and you make sure all of its methods are thread-safe, I would just store it in the application scope like this:
<cfset application.UDF = createObject("component","UDF")>
Put that in your onApplicationStart() method in Application.cfc so that it is only instantiated once. Then call it's methods from anywhere in your application as "application.UDF.getMessage()".
-Carl V.