Aria here, I'm part of the Social Care Team at Shopify. :)
I would suggest checking out these suppliers who can integrate directly with your Shopify store:
Printify fulfils from different countries including the UK. I would contact them directly at email@example.com to confirm which products ship from the UK. Streetshirts and Inkthreadable are both UK only.
I hope one of those companies will work for you. :) You can find the full list of fulfilment companies in our app store here.
Social Care Guru
We are a women fashion brand based in China, providing stylish apparel worn worldwide by thousands of sophisticated women. We pride ourselves on the amount of orders we have to the USA, Europe and Asia.
We are now looking for retailers to help us expand our business, are you interested in making some profits?
We can do 1 order dropshipping worldwide and we can offer you great discounts too.
Please email us at firstname.lastname@example.org for more details.
Here's our AliExpress website: http://www.aliexpress.com/store/708938
Tom here from Two Fifteen.
Two Fifteen are a UK based drop shipping company that specialise in the printing and the shipping of print on demand products. With over 250 products available including t-shirts, mugs, cushions and canvases. We offer worldwide shipping. All of the items we offer on the website are printed in house by our production team and shipped out within 3-5 days.
If you need any help in setting up contact email@example.com or give the guys in the office a call on 01253 302791.