<apex:page>
    <apex:outputPanel id="blockPanel">
        <c:subBlockCmp projectObject="{!projectObject}" />
    </apex:outputPanel>
</apex:page>
-----------------------------------------------

<apex:component controller="ProjectController"  language="{!LOWER(projectObject.Primary_Contact__r.Language_Selected__c)}" >
    <script>
        window.onload = function() {
            reloadLabel();
        }
    </script>
    <apex:outputPanel rendered="true">

        <apex:attribute name="projectObject" type="ProjectObject__c" description="This is the projectObject attribute for the component."/>

        <apex:actionFunction action="{!reloadLabel}" name="reloadLabel" rerender="blockPanel"/>

        <apex:outputPanel>
            <apex:inputTextarea value="{!projectObject.SampleText__c}"/>
        </apex:outputPanel>

    </apex:outputPanel>

</apex:component>

public class ProjectController{

    public ProjectObject__c projectObject;

    public ProjectController(){
        projectObject = new ProjectObject__c();
        projectObject.SampleText__c = Label.SampleText;
    }

     public void reloadLabel(){
         // Label.SampleText with the changed language attribute in the component
         projectObject.SampleText__c = Label.SampleText;
    }
}