If you want to update billing information as well on update cart button.

functions.php

remove_action('init', 'woocommerce_update_cart_action'); //Remove cart update action so that billing address will be updated
add_action('init','woocommerce_update_billing');    //Hook own function to update billing address on cart update
function woocommerce_update_billing(){
    global $woocommerce;
    if(! empty( $_POST['update_cart'] ) && $woocommerce->verify_nonce('cart')) {
       $customer_id = get_current_user_id();
       $billing_data['billing_first_name' ] =  $_POST['billing_first_name' ];
       $billing_data['billing_last_name' ] =  $_POST['billing_last_name' ];
       $billing_data['billing_phone' ] =  $_POST['billing_phone' ];
       $billing_data['billing_email' ] =  $_POST['billing_email' ];
       $billing_data['billing_cost' ] =  $_POST['billing_cost' ];
       $billing_data['billing_branch' ] =  $_POST['billing_branch' ];
       $billing_data['billing_company' ] =  $_POST['billing_company' ];
       $billing_data['billing_attention' ] =  $_POST['billing_attention' ];
       $billing_data['billing_address_1' ] =  $_POST['billing_address_1' ];
       $billing_data['billing_address_2' ] =  $_POST['billing_address_2' ];
       $billing_data['billing_city' ] =  $_POST['billing_city' ];
       $billing_data['billing_state' ] =  $_POST['billing_state' ];
       $billing_data['billing_postcode' ] =  $_POST['billing_postcode' ];
       $billing_data['billing_country' ] =  $_POST['billing_country' ];
       $billing_data['billing_date' ] =  $_POST['billing_date' ];
       $billing_data['billing_needed' ] =  $_POST['billing_needed' ];
       $billing_data['payment_method' ] =  $_POST['payment_method' ];
       $billing_data['billing_country' ] =  $_POST['billing_country' ];

       If($customer_id){
       foreach ( $billing_data as $key => $field ) {
            update_user_meta( $customer_id, $key, $field ); //Add billing data in user meta table
        }
       }
    }
}
add_action('init', 'woocommerce_update_cart_action');   //Hook update cart function again after the billing data update.