-Ideally, the estimated delivery in the Order completion email also will be showing the same date as shown on the cart page.
If this is not happening in your case, could you please share the screenshot of both of these pages? So that I can check and help.
-Shipping zone will only be considered when the address details are provided by the customer on the cart/checkout page. This is because, WooCommerce recognize the zones only when the address is provided. Hence, it won’t be considering the zone dates while showing on the product page.