This is an update to the following blog post from October 2013. http://toplessreview.blogspot.co.uk/2013/10/unknown-drama-3.html
While revisiting this query I've managed to pinpoint a very likely candidate.
Here is a collage of scenes from it.
Armed with the known clues that 1) it had a character called "Jack"; 2) there was a scene in a computer centre; 3) a French woman featured ...
... and then narrowing the timeline being considered to the vicinity of June 1987, which seemed a possibility because the clip's position in a sequence of forum posts made by the original archive clip collector (the two clips either side of the unknown were both from known programmes broadcast in that particular month)
A search of some TV records elicited a single possible candidate that fitted all the criteria. It was a one-shot drama broadcast on Sunday 21 June 1987 called Man on the Screen which went out under the banner title of Thriller (although nothing to do with the long-running 1970s ITV series of that name).
Man on the Screen featured a lead character called Jack Adams played by David Quilter. His co-star was the French actress Elisabeth Etienne, who was playing a character called Chantal. It also had a character listed as a "Computer Technician" played by Jill James.
Thriller was an ITV series made by the HTV company and appears to have only been a short series of three one-off dramas. The only other two that seemingly went out under this series title were both shown in 1986.
Because it was an ITV programme it is not covered by the BBC Genome project so instead I used the Times Digital archive to lookup that week's TV listings page and obtained the following plot details.
TV REVIEWER'S CHOICE: Man on the Screen (10:00-11:00pm, ITV). Smart Money meets Gaslight in Terence Feely's electronic thriller. With one nervous breakdown under his belt, computer genius David Quilter finds his sanity threatened once more by a couple of corrupt colleagues. Alastair Reid's mannered direction and Feely's clichéd dialogue fortunately detract little from the story.
LISTING: Man on the Screen. A computer genius is accused of a murder he cannot remember committing
Since it was about a computer genius a scene set in a computer centre fits.
And most compellingly pictures of David Quilter bear a strong resemblance to the man in the scene.
The French woman's face isn't seen all that clearly in the clip but images of Elisabeth Etienne show the same general look.
With the clues fitting all available avenues of investigation plus additional follow-ups on scenario and actor resemblance also panning out, it seems as likely as it can be (without the full programme to hand) that this query is now solved to a sufficient degree of confidence to call it.
Elisabeth Etienne's acting career was primarily in France and this seems to have been the only thing she did on UK television, although it is not shown on IMDB. She also appeared nude in the French film Femmes de personne (1984).
Man on the Screen was written by Terence Feely and was directed by Alastair Reid. It also starred Robert Urquhart, Natalie Slater, Eva Mottley and Thorley Walters.
The other two episodes in Thriller, both from March 1986, were Macho starring John Stride and Susannah York; and You'll Never See Me Again starring Leslie Phillips.