If you want to sell things on amazon.co.uk, you do not necessarily need to live in the UK, but you must be residents of the countries approved by Amazon. You can find the list of countries here. You also need to have a credit card that can be charged internationally if you are not UK residents.

You then need to apply a seller's account on amazon.co.uk.
There are two types of seller's accounts: one does not have monthly fee while the other has. You can choose either of them depends on how many items that you intend to sell each month.The account is normally approved within several minutes after application. Once the account is approved, you can upload your product inventory. Whenever customers purchase your item, you will be notified by email. You can pack and post the item to the buyers yourself, or you can use Amazon's Fulfilment service which is convenient but requires extra fees. For more about Fulfilment, you can check here. Once your payment is sent, you will get an email notification.