I don't think it was a supply issue since the RAF would later adopt with the E Wing Spitfires a pair of Browning .50s.

From what I can gather, I think it was a matter of ease of production. The .30s probably weighed less than the .50s, but given that the .30s also had quite a lot of ammunition, the difference was at best marginal.

That said, the FN series turret design by Nash and Thompson (the turrets in the rear on the Halifax, Lancaster, Beauforts and other bombers) has a variant the FN-82 that had 2 .50 Brownings instead of the dual .303s or the quad mounted .303s.

Likewise the Boulton Paul Type E turret also utilized 2 .50s. As to why it wasn't widely adopted I would gather that it just simply wasn't used as much. Given that the BP Type E was used primarily on the Avro Lincoln (a rather late war plane) and there not being many mentions of the FN-82 aside from the finalized B. VIII version of the Lancaster, I would just assume that Nash and Thompson along with Bolton Paul made more of the .303 variant turrets or that the production orders and designs didn't change until post 1943 when the Lincoln and the late B. Lancasters were being ordered.

0.50 calibre machine guns did see limited use in RAF Bombers in world war II. Mostly on the Halifax, which had some squadrons trial a different rear turret design, however that turret design did not see widespread use.

I would be speculating if I gave a reason why the .303 machine gun was a mainstay for the RAF.

After thinking about it for a bit, I don't really think that the 0.50 calibre machine gun was really that good at protecting US bombers in the skies over Europe. -The USAAF had unsustainable loss rates until long-range escorts such as the P-47, P-51 and P-38 were introduced. In addition, the Luftwaffe was very quick to change tactics once they became aware of the limitations of the heavy bombers: attack from the front.