Night fighter!
In fact WP232, the Vampire shown in the film, was a Vampire night fighter, as Nigel Patrick's character states when he arrives at 'Cairo' - it is portrayed as a Ridgefield aircraft in the movie. Hence the Comet was indeed being flown by a 'rival company'.
Egypt actually ordered 12 Vampire night fighters, but the order was blocked by the government, so the aircraft were acquired for the RAF as the Vampire NF10. When shown in the film, the aircraft was flown by 25 Sqn at RAF West Malling, remaining in service until replaced by the Meteor NF11 in 1953.
Flying early jets in the Garden of England must have been a great time to have been in the RAF; 25 Sqn later moved to Tangmere - another delightful location!