After a little bit more detective work the photo was taken at Capetown by a D.R. Gillespie as credited in the excellent book 'RMS St Helena South Atlantic Mailship' by the ship's former Radio Officer Robert Wilson. Unfortunately no date is given and the encounter is not mentioned in the text.
I am extremely doubtful now on whether the photograph was taken in 1983 as the St Helena, after her stint as mother ship for the mine clearance squadron (HMS Brecon and HMS Ledbury) immediately after the end of the Falklands War in the second half of 1982 returned to the UK but was then further chartered by the MOD for duties in the South Atlantic between Ascension, the Falklands and South Georgia in the first half of 1983 and was not in the condition in the photo during this period. She was then refitted / restored at Falmouth and finally recommenced her service to her island in late September 1983 and could not have been in Capetown after stopping at her namesake island until late October at the latest - when QE2 was definitely on North Atlantic duty (crew running in Boston marathon at that time) before heading to Germany in November for her Magrodome refit including installation of the tenders mentioned by Thomas. On looking at the photo again more closely I believe that the tenders could just be out of sight behind the lifeboats and the curve of QE"'s hull towards the stern.
QE2 did not call at Capetown on her 1984 World Cruise (Durban instead) but did on those in 1985 and 1986 before her re-engining and enlarged funnel, and I believe that it was on one of those calls that the photo was most likely taken - unless QE2 could have visited Capetown on another (non-World ) Cruise during this period?
Gary