The only Churchill tanks to serve in the Western Desert were 6 Mk III's sent out for evaluation. It was felt that the cooling system was not suitable for tropical climates. They acted as an independent unit known a "Kingforce" (o/c was Major Norris King, Royal Gloucester Hussars), attached to 1st Armoured Div. The unit saw action twice, first at Kidney Ridge then at Tel el Aqqaqir (27th Oct & 3rd Nov 1942, respectivly). Inspite of having one tank destroyed and two damaged they performed well enough for two Churchill brigades to be added to 1st Army in Tunisia. More details can be found in the Osprey Vanguard /New Vanguard books.
Also check out this site for detail photos
http://www.armourinfocus.co.uk/a22/