How to use the country set in the shipping estimate in checkout?

Hook into Mage_Checkout_CartController::estimatePostAction and set the countryId() in the billing address to the specified country.

/**
 * Use the country from the shipping estimator in the checkout
 *
 * @event controller_action_predispatch_checkout_cart_estimatePost
 * @param Varien_Event_Observer $observer
 */
public function controllerActionPredispatchCheckoutCartEstimatePost(Varien_Event_Observer $observer)
{
    $request = $observer->getControllerAction()->getRequest();
    $country = (string) $request->getParam('country_id');
    $cart = Mage::getSingleton('checkout/cart');
    $cart->getQuote()->getBillingAddress()->setCountryId($country);
}