IWM film is here :
THE CAIRO BAGHDAD SERVICE AIRMAIL [Main Title] | Imperial War Museums (iwm.org.uk)
I think the aircraft is a Vickers Vimy Commercial derivative, which had that bulbous nose unlike the original WW1 bomber. Various versions of this seem to have gone by different names (Vanguard, Victoria, Vernon etc). The latter described here :
Vickers Vernon - Wikipedia
I'm surprised it ran from Cairo. The mails brought from the UK by sea would have arrived at Alexandria, which became a key transfer point in the 1920s-30s for Imperial Airways, at the junction of their African and Asian routes, both for flying boats and landplanes. The mails would have had to be transferred on the Alex-Cairo railway, adding a day to the supposedly rush transit.
I wouldn't like to be right behind those huge, fixed-pitch props when starting the engines. Yet one of them even manages to keep his hat on !