Honestly, given the support Junta Britain had from Uncle Sam from the day of the coup until the Junta stepped down, I have to wonder if it is reasonable to say Britain lost her US Security Council permanent seat and veto at all--it does seem to me that there would never be a time when the USA would not insist on keeping it.
What I could see happening maybe is an agreement by the USA to suspend UK status on the Security Council on conditional terms. Like King Arthur it is once and future, sleeping but not dead. The deal would be agreed to in the 1990s, when the USSR had collapsed but before 9/11 when it seemed the "end of history" had been reached, by the Clinton administration. (I suspect this is something the President could do without Congress, as it does not involve new treaties, just administering the USA's role in the Security Council, which is a matter of instructions to the Presidentially appointed Ambassador to the UN. The Republicans would be screaming bloody murder about denying the Junta the effective power to veto--at the exact same time as they also sneer at the UN and mutter endlessly about abandoning it completely. But perhaps Clinton could score some cheap points with the American left, such as it is (Clinton would not be catering to them so much as moderates actually) who deplore the Junta--real leftists would be pointing out how serviceable the Junta has actually been and rolling their eyes at the hypocrisy of it all, actually. The compromise with American rightists (and to an extent European ones) would be that Britain retains its charter right, but it is in suspension at the moment.
This would require some retconning, but I think it would serve the basic narrative well. The accord government is provisionally penciled in as having its Security Council seat restored--as soon as certain conditions are met including a fixed cooling off period to see if the new UK regime is at all stable and sustainable or if it will be recaptured by the Junta (or by radical leftists). This is infuriating to British leftist radicals, but the notion of the UN SC having these permanent veto holding members is contrary to their ideals. It is infuriating to the British right, as was having their right suspended by Clinton essentially, but restoring their seat is a carrot being held out that makes more pragmatic or moderate rightists consider they are on their good behavior and must try to rein in their more extreme allies on the right, while also giving the center enough support to keep the far left in check.
Because Britain is not yet settled and the stipulated "cooling off" minimum time span has not yet elapsed, Britain has no effective SC power at the moment, just as in the canon narrative so far. It remains to be seen whether it would be activated later or not.
Realistically, the idea of the 5 veto powers was probably derived in part from Churchill's concept of the post-war order. According to his post-war memoirs of the war years, he proposed a two-tier system, in which just the Big Three would have special power at the level of the global UN organization, but each of these powers--USSR, USA, UK/Commonwealth (I am not sure if Churchill would either propose or settle for a dispersed Commonwealth system giving such dominions as Canada, Australia, New Zealand and South Africa real power presumably in proportion to population on a Commonwealth council of some kind shared with the UK itself. I presume even if he were prepared to accept such a compromise he'd prefer to make London's Parliament supreme--since Parliament technically includes the House of Lords, and Lords could be created for the Dominions, there would already be a bit of Commonwealth devolution that way I guess; I imagine Churchill would support beefing up new or revived powers for the Lords). Each superpower, with Britain presumed to continue as such, would be the supreme hegemon in a regional organization--the USA presiding over the Western hemisphere (that is, the New World, since parts of the Old World are technically in that hemisphere, including most of Great Britain, being mostly west of Greenwich Observatory!), and the UK and USSR spheres of interest having been vaguely indicated at Yalta and prior conferences, and finalized post-war. Each would be charged with seeing to it war did not actually break out in their designated bailiwick, and the ability of the global organization to interfere within these spheres would be limited severely, the global system being viewed as a clearinghouse for inter-zone relations.
Subsequently, the USA advocated for China to be included, and wartime developments led to France being granted an equal status as well. Churchill's subdivided spheres of influences lower tier organizations went by the board of course.
In practical terms, a veto is a veto even if none of the other four powers concur, so the Soviet veto initially was set against all four others, "China" being defined as KMT run Nationalist China which after 1949 was able to control only Taiwan, but retained the UN seat with the PRC excluded from the UN completely until the early 1970s, and Britain and France, when push came to shove, pretty much perforce aligned with the USA as well. Eventually switching recognition from RoC to PRC turned the SC into something more like Churchill's vision of actual great powers meeting as peers, though without the collegial cooperation Churchill hoped the Soviets might possibly offer (though he had little illusions they likely would). This still leaves Britain and France free to take fourth and fifth positions on paper, but in practice they are unlikely to oppose the USA in any grave matter anyway. Nor do numbers of SC members on one side or another of some issue matter; a veto cannot be overridden. (The Soviets withdrew their ambassador to the UN in protest of some unrelated issue when the Korean War went hot, which is why the other four members were present without Soviet check to ram through a unanimous SC resolution to support the southern Republic of Korea and make the anti-Northern alliance technically a UN mandate. Since then the Soviets and their Russian successors have been careful not to be absent from SC proceedings!) Therefore it would not make any practical difference if one of five SC veto wielding permanent members were suspended temporarily, if that one was part of the Western bloc anyway. USA and France are quite sufficient.
The alternatives to this suggestion of mine of conditionally suspending Britain's SC seat and conditionally restoring it again would be
1) abolish the whole SC veto system, which might enable the UN to operate more effectively in theory but would be likely to result in the USA and other major powers withdrawing support from it, which would probably crash it into total irrelevance--though there is a TL to be written perhaps, of some crisis resulting in the great powers repudiating the UN but then sufficiently many of the lesser powers pull together and reorganize it and fund it to have serious weight in the world versus these great powers. This is a non-starter in a TL attempting to hew as close to OTL global conditions as this one does, and pretty fanciful in almost any context.
2) permanently delete the British seat and roll forward with just 4 standing members; I think this is what the author is assuming. Functionally speaking France is a reasonable stand in for "European Community" representative anyway, and with Russia and China being two of the other three corners versus the USA, it is quite possible France, especially if they either feel morally bound or undertake formal obligations to represent the larger EC as a whole, might wind up at loggerheads with Uncle Sam. And still, if Washington is standing all alone with all three other members and the vast majority of the General Council and Secretary General all disapproving, tough noogies, the single Yankee veto still stands. The USA might wind up being expelled, or preempting this by unilaterally withdrawing.
3) other comments here assume the British seat can be transferred to some other world power.
Certainly the Islamic world might well feel badly shortchanged under the current set up and lobby for one of their nations to get the seat--but which one? On strictly Islamic terms Saudi Arabia would appear to have the inside track, but of course many of the stronger Islamic nations--Turkey, Iran, probably Pakistan, Egypt, and Indonesia--would reject that claim. So that's a deadlock nor do any of the other major Islamic nations have any sort of compelling claim. If all agreed on a single Caliph who seemed to fairly balance the claims of the diverse major Muslim nations, that might work--except the UN is not going to recognize a religious official divorced from representation of one nation-state. If Ataturk had settled for a stringently restricted Sultanate with real power going to a parliament equivalent to the OTL republic (which I gather he controlled as a strongman) then today maybe a Caliph named by the Ottoman Sultanate might in future be nominated so as to balance the interests of most Islamic nations, but surely Shi'ite Iran would dissent since the whole Sunni/Shi'ite split is a dispute about which persons were the historic rightly guided Caliphs anyway. Without PODs going back a hundred years or more, or something really revolutionary happening in the future, an Islamic nation on the permanent SC is a pipe dream unless some such nation muscles its way into superpower status in general.
Realistically there is no nation in the world that can claim the sorts of far-reaching power the USA can still project, though perhaps our ability to retain that status is in doubt in the near future. No other nation seems close to claiming the same sort of power; the PRC which is already seated seems closest to it. India is recently striving for greater hard power, but like the USA I have the impression it is happening at the expense of eroding her soft power, so that an Indian claim to an SC seat would have been stronger in the Cold War era as leader and most meaningfully nonaligned member of the "nonaligned bloc."
India also might somewhat perversely claim some sort of vague claim on the British slot due to her forced marriage under the British Raj but that seems pretty silly.
For reasons similar to why a collectively appointed Islamic bloc seat is preposterous, I can't see any movement to grant the Commonwealth collectively the power to fill the formerly British seat. This might actually work to be sure, with Canada, Australia and New Zealand presiding over a bloc of smaller poorer former colonies that have not yet repudiated their Commonwealth membership to work out a genuinely multinational and balanced procedure to name and instruct such a generic Commonwealth ambassador, but the simple fact we would not have one person representing one nation would torpedo any acceptance I fear.
There is no precedent for handing over one of these seats to a different nation. The switch from Taipei to Beijing was technically, according to claims of both Chinese governments at the time, a matter of switching recognition of which government actually spoke for China. China as an abstract nation never lost its seat.
There is no compelling candidate to be the "new Britain" or even predictably the "new China."
Having offered as IMHO the most sensible options as being either that Britain's SC veto holding membership was never in doubt or abeyance despite offended liberal sensibilities (really, this seems the most probable if somewhat appalling outcome) versus compromising with a "suspended membership" which is admittedly also unprecedented, I suppose the least crazy third option is to just assert the UN agreed to eliminate the fifth seat completely.