The link here takes you to a PhD thesis on the history of the New Zealand SAS. Definitely worth a read, and it helps debunk the "British SAS seconded to Australian/New Zealand SAS" myth.
The period of time that it covers was during my time of service.
I knew at least 95% of the NZ personnel mentioned, a lot of whom are no longer on this planet.
It is a fair and accurate dissertation of the events that occurred during that period concerning the NZ SAS.
The only mention that I saw of Brits serving in VN came during the 3rd rotation of 4 troop in so much the Aussie 2IC was ex Brit special forces.
Edit to add
There was mention made of an Aussie trooper falling from a quick extraction rope.
His body was never recovered at the time and sometime in the last 5 years or so, his remains were recovered and repatriated to Australia.