I just found out that the value property is ignored when the attribute property is defined. Although this does make sense it is not documented. In other cases (e. g. CDataColumn) an exception is thrown when two excluding properties are defined. I think this could be the way to go here as well.
Example:
<?php
$this->widget('zii.widgets.jui.CJuiDatePicker', array(
'name'=>'Model[start]',
// 'attribute' => 'start', // Needs to be commented out for the value property to work
'model'=>$model,
'value' => date("d.m.Y",$model->start), // attribute may not be defined to use value
'language'=>Yii::app()->language,
));
?>