Hughie, the NSW Registry of Births, Deaths and Marriages have an online Historical Index Search - only an Index but every record can be purchased by clicking 'Buy Now'.
Ancestry does not allow us the put the website url in messages. Google the NSW Registry if you have not already found the website. It has lots of info about searching family history and what info is usually on certificates. We can use a credit card here but I do not know if you can use a card from the UK.
Info from one of the webpages.......
Purchase online (If you supply the Registration Number, you receive a discounted fee as the Registry does not have to search for the record.)
If you are not able to apply online, you have the option to download, print out and complete a: Family History Certificate Application Form
To purchase quote the "Name and Registration number 13944/1965".
Hope this helps.