If you store inventory in a warehouse in a European country that is not your business' home country, you have to register straight away, i.e. your VAT threshold becomes zero. So if you are a US, or non-UK European business and you ship goods from the UK to your customers (no matter where they are) you need to register for VAT.
The exception is if you are shipping goods from your business' home country. For example for a UK company, there is a threshold of £85,000 (looking back 12 calendar months), below which you do not need to register for VAT (subject to not having exceeded any of the Distance Selling thresholds.