Bill Gunston's 2009 book Nimrod: The Centenarian Aircraft doesn't include much about that plane's service during the 1982 Falklands War. On pp96-97, Gunston wrote that the upgrade from Nimrod MR.1 to MR.2 was about half done when Argentina invaded, and thereafter a rushed effort was made to add flight-refueling probes to MR.2s, for maritime reconnaissance in the South Atlantic to support Britain's Operation Corporate. But "whether MR.2 Nimrods played a direct role in Operation Corporate has not been revealed". The three Nimrod R.1 SIGINT spy planes also hurriedly got FR probes. Gunston wrote (p107) that at least one R.1 "was able to play an active role in the [Falklands War]. After the conflict, No. 51 Sqn proudly added 'South Atlantic' to the battle honors on its standard, but how it operated, and from which airfield(s), has yet to be disclosed".
Under continual attack by Argentine warplanes, the UK's Task Force came to rue that it had no airborne early warning. Yet sending one of the prototype Nimrod AEW.3s to the war zone seems to have been judged unworkable.
In his bestselling memoir, Admiral Sandy Woodward affirmed (pp289-291) that Nimrod MR.2s appeared over his task force, but he panned their effectiveness. "My basic annoyance with the Light Blue [i.e. Royal Air Force] involved their expeditions down to the war zone in the big RAF Nimrod maritime patrol aircraft with its excellent Searchwater radar. The idea had always been that they would fly over the area at a suitable height and report anything they found to us. This they did with whole heart, tireless effort and great enthusiasm. The trouble was they kept getting it wrong..." Woodward gave some examples of very mistaken surface contact reports from patrolling Nimrods. He was incited to send a rude signal of complaint to London, "indeed it set a new benchmark for rudeness". In its response HQ "informed me coldly that I appeared to have upset the entire High Command of the Royal Air Force. My reply was impolite. I intimated that I did not give a **** how upset they were, they had better start concentrating on accurate factual reporting".
In his 1992 book Sea Harrier Over the Falklands: A Maverick at War, Fleet Air Arm squadron leader Nigel 'Sharkey' Ward contradicts Admiral Woodward but evinces similar disdain for Nimrod MR.2s and for the RAF's efforts (p218): "The Nimrod was never once in evidence over the Task [Force] or in adjoining areas. If it had been it would, like the Vulcan, have been clamouring for fighter cover... The nearest the Nimrod came to the scene of action during the fighting was 600 miles to the north—and that was a one-off occasion". After the war as the RAF massaged the record (p270) "even the Nimrod with its Sidewinder fit was passed off as a genuine threat to Argentine fighters, and the papers were written in a manner which suggested that the Nimrod had actually appeared in the skies over the Task Force and the islands—which it did not".
Richard J. Aldrich mentions in his book GCHQ (rev 2019 edition, p390) that during the Falklands War, "The poor performance of the Nimrods, on which Britain had blown such a large slice of its sigint budget, is still a mystery. Lt Colonel David Chaundler, who served in the Defence Intelligence Staff until halfway through the campaign, asserted 'The Nimrods were achieving very little', despite the enormous effort that was being put into refueling them. At least eight maritime reconnaissance Nimrods operated from Ascension Island, and one R1 sigint aircraft seems to have flown from the Chilean base at Punta Arenas".
Aldrich's last sentence above cites Jon Lake's article in the July 2001 issue of Air International, "Nimrod R.1: The RAF's SIGINT Platform Extraordinaire". Lake wrote (p31) regarding the single Nimrod R.1 sent south for the Falklands War that "the operating base used by XW664 remains secret, but it is believed that it left [RAF Wyton in Cambridgeshire] on May 5, transiting via Bermuda... It has been rumoured that the [R.1] operated from Ascension Island, or even from Punta Arenas in Chile, perhaps alongside a detachment of 39 Squadron Canberra PR.9s, but this is pure speculation and cannot be confirmed". For his 2009 book Bill Gunston either overlooked Jon Lake's article or (unlike Richard J. Aldrich) considered its Chile speculation doubtful.
In his 2014 book Exocet Falklands: The Untold Story of Special Forces Operations, Ewen Southby-Tailyour claimed (pp173-174) that Nimrod R.1 XW664, backed by a VC10 tanker, made three flights from Chile's San Félix Island in May 1982, staying within Chilean airspace. It would have flown more as was requested by Admiral Sandy Woodward, but the reveal of Operation Plum Duff's landed Sea King helicopter made Chile get cold feet, and XW664 returned to RAF Wyton on 22 May. Southby-Tailyour provided no source for the above.
Perhaps Chris Gibson's forthcoming new book Hawker Siddeley HS.801 Nimrod: Outstanding in the Role, intended as a definitive technical history, will provide fuller info about Nimrod service during the Falklands campaign. With Nimrods retired fifteen years ago (and Chile now openly backing Argentina's claim to the Falklands, so no need to continue shielding them), maybe His Majesty's Government has since declassified formerly secret matters, allowing richer detail. A representative at Crécy Publishing tells me that publication of Gibson's book has been pushed back to October of this year. I look forward to it.