Head to this form: https://amazon.com/podcasters
On the next drop down, select how many podcasts you're submitting.
Note: This is NOT the amount of Episodes your podcast has. This is whole Podcasts.
On the next screen, enter the name of your Podcast, and your Podcast.co RSS Feed.
Here's a bit more info on where to find your RSS Feed URL: Where Is My RSS Feed?
On this final screen, enter your name and the email that is on your RSS feed. In most cases, this will be whatever email you use to log in to Podcast.co.
For Hosting Provider, scroll down and select Other, then type "Podcast.co"
Select the main target audience country for your podcast. This won't restrict your podcast from other countries, but just gives Amazon an idea of where your audience is coming from.
Make sure you accept their terms and you're all done! Your podcast will be live in 24 hours at: https://music.amazon.co.uk/podcasts