I need to propose the network set up for a company with four offices in the UK (Birmingham, Edinburgh, Cardiff and London), one office each in Hong Kong, Sydney and New York .I need to select the appropriate network technique and also select appropriate provider(s) for the internet connectivity.
Can anyone advise on how to approach this?

Yes. Higher a consultant who has experience in global intranets and security. He will start by evaluating your needs, then design the best solution.

It sounds as if this may be a school/course project/test. Is this the case?

Why were you asked to provide a Network/Communications solution of which you understand very little?