Instructions:
Log into amazon.com with your current amazon.co.uk account.
Order it as you usually would on .co.uk
When you get to the end of the checkout it will be greyed out and you won't be able to complete the order (due to UK address)
Click on the billing address, bottom right
Find a valid zip code and city in the state of your choice that don't charge sales tax! (Alaska, Delaware, Oregon)
Make up the rest of the address and phone number
Complete the sale
DON'T click on Amazon's "Download Now" button. Wait for the steam code to be emailed to you or click on the "Product key only" link under the "Download Now" button.
Register code with Steam
(credit to majnu on AVForums for the instructions)
Personally that looks over complicated to me.
Just add an american address against your credit/debit card in your 1-click settings.
Note use an address from a state that doesnt incur extra tax. I use a hotel address from Alaska.
In your amazon account is a link to "games and software". You can download games from here, and view your "product keys" to enter in to steam, origin etc.