The voice reading the selection from Winckelmann’s account of the Herculaneum papyri at about the 4 minute mark is that of Otto Skutsch (University College London):
The moving force behind the production of the film was Eric G. Turner (University College London), who was also featured a couple of times on screen:
In addition, we also see Michael Haslam (emeritus professor, ULCA, right), with Eric Turner (center) and a still unidentified person on the left (UPDATE 17 March 2019: Helen Cockle identifies the man on the left as Alan Griffiths):
And finally, seen here taking a break from arranging papyrus fragments in order to comb his moustache, Walter Cockle (University College London):
If anyone has other identifications, please let me know in the comments.
[UPDATE 17 March 2019: Please see the comment below from Helen Cockle for further identifications.]