R – Update Panel inside of a UserControl inside of a Repeater inside of another UpdatePanel

Since you're using ASP.NET AJAX, you'll have access to a pageLoad event handler, that gets called each time the page posts back, be it full or partial from an UpdatePanel. You just need to put the function in to your page, no hooking up is required.

function pageLoad(sender, args)
{
   if (args.get_isPartialLoad())
   {
       //Specific code for partial postbacks can go in here.
   }
}

Have you thought about explicitly setting an asp:AsyncPostBackTrigger with the btn1 control in the up1 UpdatePanel control.

<asp:UpdatePanel ID="up1" runat="server">
    <Triggers>
        <asp:AsyncPostBackTrigger ControlID="btn1" EventName="Click" />
    </Triggers>
    <ContentTemplate>     
        <asp:Button ID="btn1" runat="server" Text="Sample Button" />  
        <asp:Label ID="lblTime" runat="server"></asp:Label>         
        <cc1:MyCustomControl ID="MyCustomControl1" runat="server" />                 
    </ContentTemplate>
</asp:UpdatePanel>

Edit: How you tried to explicitly call the Update method in the button's OnClick event for the Update Panel? This includes the Update panels embedded within the custom control.