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You need to click on "get eos" on the main eosio page and then tick all the boxes in the popup window and then click continue. This will take you to the token distribution page. On this page, towards the bottom, you will find the instructions on how to register your tokens

That button doesn't work "The EOS Token Distribution is restricted in your region" and I'm from the UK.

You are correct - they restrict access to visitors from UK, US and China. The instructions above will not work for you!
You simply can not access those pages.
I'm working on new instructions for US folks.

Do you know of a way of registering them using a BTS or STM key for US users?

Get a trial account at VirtualShield (VPN), and IP masquerade as from another part of the world. Places like USA, Canada, UK, even Singapore will not work.