Show shipping rate on Cart screen in Craft Commerce
I've not tried this so early in the checkout process, but how about adding the following to your Add To Cart forms? Replace the # with the correct value obviously.

&lt;input type="hidden" name="shippingAddress[countryId]" value="#"&gt;
&lt;input type="hidden" name="shippingMethod" value="#"&gt;


That way, as soon as people arrive on the Cart page a country has been added already as a shipping address, which, I think, should be enough for shipping to be calculated.