If you used the BBC's recording, you'd probably need to obtain sample clearance if it were to be used commercially or for public performance to avoid copyright issues.
I suspect the original of that sound, many years ago, was probably created using an oscillator and simply turning a frequency pot to get the pitch swoop required, plus some e.q. and maybe compression.
The late Delia Derbyshire and the BBC's Radiophonic Workshop did an amazing job on Ron Grainer's original score for that theme, and turned it from an r'n'r/boogie kind of thing into an electronic masterpiece. All done with very basic equipment and a great deal of old-fashioned sticky-tape and razor blade tape editing. Often those edits were done a note at a time.
And I still think that the early version of the theme is the best :-)
Nowadays there are loads of ways to create that kind of sound. As well as the other suggestions here, I'd try oscillator sweeps (just turn the knobs) and resonant filter sweeps as well, adding some LFO to add a bit of vibrato or maybe some FM of the filter frequency. It's easier if you're not restricted to the stepping between notes of MIDI - if that's a problem, try using glissando to smooth things out.
Or get a voltage controlled analogue synth :-)