Hank here from Shopify.
Shipping can certainly be a complicated issue at times, but once you set your profile and rates up correctly, you can be confident knowing that you should not run into any issues with your store.
The way you have described this problem you are facing sounds like either the drop-shipping app you are using is not added to your rates (if you are using the drop-shipping model), or, there are no conditions set up for your rates. It could also be a case that the conditions are not applying to the country you wish them to be, in this instance, the U.K.
The first thing you need to do is to check your shipping profiles by going to Settings > Shipping > Shipping profile > Manage profile. In there you can create different rates for your products and create either price-based rates or weight-based rates. These are known as flat shipping rates.
Once you select your region, ensure you have conditions set up to match your rates. From my below test store, you can see that customers that spend $10, receive free shipping, and this is specific to the U.K only.
Let me know how you get on with this, or if you have any other questions for me!
All the best,