Normal British infantry wore the Brodie helmet, the black Beret was restricted to the Royal tank regiment (later Royal Armored Corps) and the red beret was the mark of the parachute regiment. In combat, the paras actually wore a steel helmet of a different design (more similar in appearance to the US helmet) but tankers did wear their berets in combat.
The utility of a small compact headgear with no brim (easier to look out viewports or into gunsights etc.) inside a tank was the reason for the adoption of the beret between the wars. I believe infantry serving alongside tanks usually wore helmets as usual. There may also have been different rules for officers and headgear in the British army as well, but perhaps someone else can comment on this. General (later Field Marshall) Montgomery in particular is known for wearing a Royal Tank Regiment beret although he was not typically under fire.

Your question reminded me of this passage from Antony Beevor's "D-day : The battle for Normandy"

"Lord Lovat's 1st Special Service Brigade also landed near Colleville. His commandos had thrown away their helmets at the last moment and wore their green berets instead, with their own regimental cap badges."

The 1st were in the second wave to hit Sword beach on 6th June 1944. Beevor goes on to say that Lovat's personal piper played the bagpipes as the men came ashore.

So it definitely did happen, but how often and how widespread it was I couldn't say. The commandos were (and I understand still are) very proud of being able to wear their green beret, so I assume that has something to do with the practice (at least in this case).