Is the Doctor Who stage play available for production?
The most recent rights-users were Big Finish Audio. They produced an audio-drama version of the play in 2008 which credited BBC Worldwide as the rights-holders. Similarly, a 1989 audio-play version produced for the BBC also credits the BBC as the mark-holders and BBC Worldwide as the rights-holders. I see no good reason why this shouldn't continue to be the case.
The original writer (Terrance Dicks) & producer Mark Furness, who apparently paid the princely sum of £500,000 for the rights to use the Doctor Who, Daleks and Cybermen characters (as well as various trademarks such as the Tardis) may also have some say over whether and when their creation can be used.
Furness negotiated rights from the BBC, and found the costly £500,000 budget. The dazzling on-stage special effects, involving both lasers and computer graphics, came from Foys of Las Vegas who, among their many stage productions, had produced the simulated space walks for NASA. Rehearsals began 27 February 1989 for a Thursday, 23 March opening date in Wimbledon Theatre.
Doctor Who - The Ultimate Adventure By Mark Olsen
There is currently no legal version of the script available to purchase but there is a transcript of the audio-play available online here