There are a few things to consider, which hopefully will get you started on your journey.

The first is the VAT situation, if you sell into the UK now Amazon will remove the VAT due and pay it to the government, its the same in any EU country and the distance selling thresholds I believe no longer apply to non dom sellers.

The second, is around VAT registration. If you hold any stock on UK soil its the same as EU countries you then have to hold a VAT number and submit returns. The same applies to EU countries. Even when Amazon remit the VAT, if you have stock in a fulfilment or prep centre you have to be VAT registered.

If your using something like Pan European and your stock is held across multiple countries or moved, then unless you opt out, each country the stock is in will require a VAT number.

Then it depends on the items your selling. If you sell anything battery related, batteries or containing batteries, then unless already registered under the brand you will be responsible for registering with the Environment agency and fulfilling the WEEE obligations and reporting and takeback scheme fees.

For plastic packaging there are thresholds, where if you import a certain amount, you have to register.

For any products you are selling that are your own or branded by you, you will need to complete a declaration of Conformity and ensure they meet the safety standards for that product.
If your supplier is the brand, then you will need to make sure you can get the DoC of them.

If your listing a product not already sold, then you will need an EAN/Barcode. Amazon only recognises GS1 as the official barcodes, any barcodes outside of this can and eventually are flagged and removed.

If your planning on selling an established brand, its best to get a letter from them authorising your sale on the country concerned. Or you could run into parallel import issues as some brands have exclusive distribution with sellers.

Also always remember, unless purchased from a wholesaler or manufacturer Amazon will consider your items as used. So if you were for example to do Arbitrage, and buy some makeup sets for resale.
Even though they maybe brand new in the box, if you don’t have a wholesale invoice, amazon may see them as Used because of their policy.

For moving goods in and around the EU and UK you may find intrastat requirements.

You will also need an EORI number to do this and broker to move the items in.

Hopefully this helps and if anything is wrong I am sure someone will correct me