Of course the Government of the United Kingdom does use (and I emphasise use as it doesn't really have) a coat of arms:
This is a variant of the Royal coat of arms, which is used quite a lot by the government, for example on the front of the British passport (which is technically government property):
and in the logos of government and state departments:
But while the government uses these arms it is not the owner of them. As stated before they are a variant of the Royal coat of arms, and as the government is "His/Her Majesties' Government" and constitutionally gets its power and authority from the crown, it uses a variation of the Royal Arms. Now I am not disputing that nor saying the UK Government isn't run by the people for the people, nor am I saying it should have a coat of arms. This was really just a heraldic exercise. For example if the Government is granted its own coat of arms by the College of Arms for its own use, what might it look like. My idea is below:
Firstly the shield: The shield is divided into two colours green and red, these are the colours of the house of Commons and the House of Lords (like most democracies Parliament has a senior and junior chamber for example broadly speaking House of Representatives is the US equivalent of the House of Commons and the Senate is the US equivalent of the House of Lords)
The crowned portcullis is the symbol of the Palace of Westminster where Parliament has met since Tudor times at least. According to Wikipedia the Portcullis was common on the arms of English nobility this was adopted by the Tudors for Parliament (adding the crown to symbolise their sovereignty).
The Stars can have duel representation they can represent both the Constituent countries/Home Nations that make up the United Kingdom, or they can represent the three devolved governments within the UK (Scottish Parliament, Welsh Assembly and Northern Ireland Assembly (each of which also has a logo/heraldic badge)) which get their power (and funding) from the central British Government, while the fourth represents the local governments of UK dependencies and overseas territories, which also have devolved powers to run local affairs.
The Royal Arms of the three historic Kingdoms that are present in the Royal Arms.
Secondly the supporters: the two monstrosities that are crossed are ceremonial maces. Each house of Parliament has one which is displayed in the chamber when the relevant house is meeting, it basically represents the sovereignty of the monarchy in the absence of the monarch. They are also sort of unofficial symbols of the individual house. I think as well as symbolising government of the people, for the people by the people it strikes a good balance as it also represents the sovereignty of the sovereign, via the Royal arms crown, garter ect.
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