Shortly after I made the above post I decided to try sending an email to the British Embassy, and copying in the relevant people at New Dynamic Institute.

The very next day the money was transferred to my bank account in the UK!!

I'm not suggesting the British Embassy did anything to help me - their reply stated they don't get involved in commercial disputes, and they gave me a link to a list of English-speaking lawyers in China - but just the fact of sending the email, and making NDI aware that it'd been sent, seems to have persuaded the school to pay up.

I understand non-payment at the end of a contract is quite a common problem with language schools in China. So if anyone else has the same problem perhaps they could try the method I used.

BTW The branch involved is in Shijiazhuang in Hebei province, and the member of the HR staff who dealt with the matter is called Mandy Wang.