TL;DR: Yes, in order to earn commission from international sales, you will need to have signed up for the Amazon Associates program for that country. Check out our sign up guide for a walk through!
The various Amazon Associates programs are country-specific, so your Amazon US ID can't be applied to UK or Canada sales.
Our 5-step translation process will always redirect an international shopper to their local Amazon storefront when we're able to find a product match, regardless of whether or not you have an Associate ID for that region.
However, you can not receive commissions from sales made in regions where you don't have an Associate ID, so you'll need to apply to each program separately.
As much as we'd love to take care of this for you, we're not allowed to apply for the Amazon Associates program on your behalf. It may seem a bit daunting at first, but the forms for each region are almost identical and once you're finished with this you'll be all set to collect international commissions!
When you're ready to begin, check out our Amazon Associates Program Guide for help getting signed up!
As a side note: Once you're signed up with Amazon UK, applying to other programs in Europe (Germany, France, Spain & Italy) only requires a few clicks, so that's the UK is a great place to start.