Uk VAT for non-Eu business

I am running an online business through Amazon and I have many customers from UK. My business is registered in Hong Kong and running through as well. My question here is that I receive orders from UK customers and we do not charge any UK VAT on them and sometime customer request for VAT invoice and we tell them that we are not VAT register business as we are out of UK VAT scope.

Recently, we are planning to hire another company to ship our orders to our UK customers within UK. We will send them goods from Hong Kong and also give them customer orders details. What they will do it is simply read customers’ orders and pick the goods, pack them and send to customers.

UK company is totally an independent company which provide same fulfilment services as amazon do but relatively cheap prices. My question is now. If we use this company to send our goods to customers because it will save time and customer will receive their orders more quickly.

So do we need to register for UK VAT and pay UK VAT?

Thanks and waiting for some helpful advice.