Interesting thread on a quite technical issue.
For fighters with fixed guns upto about 1940 were to be set to what came to be called "The Dowding Spread".
The press of war saw many such Squadrons, on their own inititiative, set a tighter pattern ("point harmonisation") at closer range.
The range chosen was related to the likely target on ops: eg day vs night interception.
The work was carried out by the Armourers, not the pilots, though in the case of "august" figures, at their direction, apparently, as to range.
It's unfortunate that these developments seem to have dominated later historical discussion & summary.
The argument, basically, was to increase the chance of at least some hits, esp for those pilots not quite such good shots.
From the bomber crew's point of view, early in the war the single front and turret Vickers (or still, sometimes Lewis) of the Blenheim and the Battle for example needed (apart from regular cleaning & maintenance) only a check on sight alignment, at some interval. No details to hand. When the twin-VGO Blenheim turret came along, so too came the need for gun harmonising.
The OP asked "how often an air gunner would have to perform the task".
I can offer one example on ops, for a Blenheim IV crew in late January 1942.
At that time, 211 Squadron was working up at Helwan to readiness for immediate posting to the Far East theatre, 24 aircraft by air, ground staff etc by sea.
Sgt JB Keeping was WOp/AG in the crew of P/O B West RAAF (pilot) and P/O G Ritchie (Observer) and their aircraft at the time was Z7699.
Keeping kept an almost daily diary of his doings from the start of 1942. It might be of interest to record his remarks of the time on harmonising.
"Monday 26 Australia Day [Jan 42]
At 0900 I went down to 7699 and tried to harmonize my guns [the twin machine guns in the rear turret]. After spending all my time it was lunch time. Mal and I went to Main Camp Mess after which we returned and spent the rest of the day [sic: at the aircraft, presumably]. Some mail arrived from Gretta and George. At night Mal and I went to the pictures.
Tuesday 27 [Jan 42]
Rose at 0700. After having breakfast a truck called for us and our flight [The crews of West, Harrison and Paterson] went into AHQ to be briefed. After briefing Mal and I had lunch and went around to AIF canteen and purchased some things after which we came back to Helwan and went straight to our kite. After harmonizing my guns, checking my ammo and TR9 it was time for dinner. After dinner I wrote."
At that time, Keeping, having completed his 70 OTU Blenheim Ops course in the last qtr of 1941, had been posted to 211 Squadron on 28 Dec 41, shortly after their brisk extraction from 72 OTU duty. The Squadron's 24 aircraft left Helwan in stages over 4 days from late Jan 1942: West, Ritchie and Keeping in Z7699 departing as the leading aircraft of the second section of three aircraft departing on 28 January. Thereafter the only mention of his attention to the guns was the need to clean them, on Fri 6 Feb after arrival at P2, Sumatra.
Sgt John Blane Keeping RAAF 404295 was killed in action on a night raid 10/11 Feb 1942 over Kluang, Malaya, in Z7699 with S/Ldr KCVD Dundas DFC and P/O GR Ritchie 400543 RAAF.
Sources and further reference
KCVD Dundas DFC