There is a Next site which delivers (free) in 4 days. www.nextdirectory.ae it is really good but doesn't have exactly the same bits as the actual Next site.
Does next here in Dubai have a website, or can I get here what is on the next direct UAE website in store?
I think you have to use the next direct one, I haven't seen Next Dubai online here
http://us.nextdirect.com/en/
Probably be cheaper too Polly