update_order_review( ) on button click

All you need to do is call a trigger on the body to update the cart.

$( 'body' ).trigger( 'update_checkout' );

This will automatically call all the subsequent AJAX calls needed to refresh the cart information, including the order review.

In Checkout page:

jQuery(document.body).trigger("update_checkout")

In Cart page:

jQuery(document.body).trigger("wc_update_cart");