As others have mentioned, there was an AA mount for Bren on early British tanks and Robert-w's description of it being like an Anglepoise lamp is bang on. The Lakeman AA mount was usually used with a Bren gun with a drum magazine Engines of the Red Army in WW2 - Bren Gun Mk.1 Light Machine Gun on Lakeman AA mount has some pictures and some modelled shots. It describes it as follows: . If you do further searching under bren aa mount or Lakeman mount, more images and discussion shows up. The overall conclusion is it was overly complicated and more trouble than it was worth so was soon discarded or 'lost' by tank crews.
It seems quite possible that in 1941 in the desert there was a need for mounts but by 1944, with allied forces in ascendancy and the provision of purpose-built AA Crusader tanks and more AA artillery, that they didn't think the tanks would need such a mount. P.S. the one piece of militaria I do own is a box for 100-round Bren gun magazines; I 'inherited' it from the former owners - who were veterans - of a cabin that my parents own. Though I have no idea where they served, or where they would have picked it up.
Found a mention of a mounting on the White Scout Car. But that had an American .50, right? Did these use a Besa in British service or the .50? In response to a survey on SPs, question about what sort of MG they wanted, one AT regiment replied: "The Besa fitted in mounting similar to the White Scout Car. Failing that one Bren per SP with 100 rounds magazine. The present issue of a rifle is not satisfactory as difficult to store in the SP."
The headache of sorting 7.92mm for the Besa, .30-06 for the Browning M1919 and .303 for the Brens/Rifles! Or is itjust hedging bets they'll alway have at least one type?