The newly set value is not shown. Instead the default text is displayed.

As a workaround you can set the TimeSpanWatermarkTemplate property.
 <DataTemplate>
	<telerik:RadTimeSpanPicker>
		<telerik:RadTimeSpanPicker.TimeSpanWatermarkTemplate>
			<DataTemplate>
				<TextBlock Text="Some text here" />
			</DataTemplate>
		</telerik:RadTimeSpanPicker.TimeSpanWatermarkTemplate>
	</telerik:RadTimeSpanPicker>
</DataTemplate>