Aria here, I'm on the Social Care team at Shopify.
Without seeing your products, it's difficult to advise you but it sounds like a 3D printing company could help. Shapeways Fulfillment has an app in our app store. You can find examples of necklaces they've created and the materials they have available on their website.
Orders shipping to the United States (excluding Puerto Rico), Canada, Australia, and New Zealand, ship from their distribution centre in Long Island City, New York.
Orders shipping to the rest of the world ship from their distribution centre in Eindhoven, Netherlands.
You might have been hoping for a company who can ship from the UK. This business model is still pretty new, there aren't a lot of integrated options out there but it may be worth searching for a 3D printing company in the UK and ask if they are willing to do print on demand. You can set up a custom fulfilment option for your store.
Since you want to customize the necklaces, you could combine this with an app such as Customify to let your customers specify their requirements.
Do you think this would work? If you have any requirements I have missed, please let me know.