I looked and BBC did do some documentaries in between 1995 and 2001.
"Make it So" A look at the special effects, costumes and make-up of the franchise
'MAKE IT SO' Directed by Susan Kemp, Produced by BBC Scotland, Length: 30 mins., 2001
Rubber aliens, mini-skirts and home-made special effects are what we really remember from 35 years of Star Trek. The programme goes behind the scenes to ask just how the tricks are done and commemorates photon torpedoes, tricorders and the 24th century bra.
"The Warp Factory" How real science influences Star Trek, and vice versa
'THE WARP FACTORY' directed by Elaine SHEPHERD, Produced by BBC Scotland. 30 minutes., 2001
source source 2
From the phasers, transporters and warp engines of the original series to the plasma conduits and microfusion initiators of 24th century Starfleet, Star Trek writers have plundered all their best ideas from science. Having created four ever more advanced space-ships - where is there left for Star Trek to go?
TV documentary. How STAR TREK writers have combined science fact and fiction to realise fantastic technological futures, looking at such aspects of the series as warp drives, artificial gravity, tricorders, transporters, and how the actors coped with the scientific jargon in the scripts. With contributions from science presenter Gareth Jones, Andre Bormanis (science advisor on NEXT GENERATION, DEEP SPACE NINE and VOYAGER), former astronaut Buzz Aldrin, comedian Sean Cullen, journalists Jay Rayner and Stuart Maconie, Bob Justman (co-producer on the original series and NEXT GENERATION), Ron Moore (executive producer NEXT GENERATION and VOYAGER), Rick Berman (executive producer NEXT GENERATION, DEEP SPACE NINE and VOYAGER), science writer Dan Cray, Brannon Braga (executive producer NEXT GENERATION and VOYAGER), and actors Nichelle Nichols, Majel Barrett Roddenberry, LeVar Burton, Marina Sirtis, Patrick Stewart, Kate Mulgrew and Robert Duncan McNeil.
"Horizon - The Science of Star Trek" Roger Penrose, Stephen Hawking and Lawrence M.
Krauss discuss the real science behind the Treknobabble. 48 min, hosted by André Bormanis. 1995
"The Star Trek Story" Documentary exploring the history and cultural impact of the series
that is over 2 hours of documentaries that Paramount would have to license from the BBC.
Perhaps they will include
2-hour Star Trek: Beyond the Final Frontier
2007 History Channel special - that were included on the DVD side of the HD-DVD combo release on TOS season 2 or 3 Blu-ray or else on the special edition of this.