It would be rather awkward for the courts to argue that life terms were inconsistent with republican government, given that many of them, including the highest court in the land, serve life terms themselves. What’s sauce for the judicial branch is sauce for the gander.
A lifetime appointment for a judge is very different than one for an executive.
Also, the governor of North Carolina is a woman now. That was actually what alerted me to this being a zombie. 
What about lifetime appointment for a senator?
Given the time period in question, you really should say “Damn your eyes, Sir!” 
You are quite mistaken, sir. Great Britain to this day does not have separation of powers. Parliament legislates, administers and adjudicates. All in the name of the Queen, who is the font of authority to do all those things.
Did someone change the Constitution?
Separation of powers is not a necessary characteristic of a republic as the term is and has been used in the United States.
Home Secretary is essentially a Minister of the Interior, with responsibility for police, immigration, internal security and the like.
Essentially the Secretary of State for Justice. He was also until recently the head of the judiciary.
The Cabinet Office is the co-ordinating office for government activities - sort-of-but-not-quite like the White House Executive Office, but for the whole Cabinet rather than just the PM.
Chancellor of the Duchy of Lancaster is a sinecure - i.e. there are no formal functions related to it (beyond being responsible for the Duchy of Lancaster to Parliament) - think of it as a minister without portfolio: can be given miscellaneous government tasks and through the office has a seat in Cabinet.
The Captain of the Honourable Corps of the Gentleman at Arms is the Government’s Chief Whip in the House of Lords; so he attends Cabinet meetings to learn what to enforce in the Upper House, but has no vote at such meetings.
There is no real separation of powers today. At that time there indeed was, Parliament had chopped a Kings head off to make that the case.
The fact that the executive authority of the Government must have a majority in the House of Commons (and therefore practically control the legislature) is something which is a product of the 19th century, specifically Sir Robert Peel’s failed attempt to rule without such a majority in the 1830’s. Its a convention to this day in the United Kingdom (Great Britain as a nation state having been abolished in 1801). At that time it was not unknown for the executive not to control Parliament (and indeed William Pitt would lead a government without such a majority). There is nothing stopping the Queen from appointing some random Belfast tamp as P.M and he could quite properly exersize all the powers of the office without ever setting foot in Parliament. Not going to last long, but the point remains.
I can assure you that checks and balances do exist in the British system, but perhaps not in the same manner as in the US. Parliamentary Committees, especially, are extremely wary of breaching the separation of powers and trying to force a policy on the government - they strongly consider the government should have freedom here, and the House alone has the power to force a policy on them. Parliament and the judiciary are also strictly separated, and do their best to keep out of the way of each other.
The Executive and the Legislature also divide powers between them. The situation is that the Government has an inherent advantage in legislation by virtue of its elected majority, but it is up to Parliament to inquire and scrutinise and pick holes in things to implore the government to improve its own work, and if necessary to compel it.
So I would argue there is a separation of powers in the UK, but of a different nature but also to a different extent. It’s true there is an overlap of powers, a ‘fusion’ of powers, as it has been labelled - but to argue there is no separation of powers is haphazard.
Have to disagree, DSYoungEsq.
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Parliament legislates on its own authority, not in the name of the Queen. The Queen is a component of the Parliament, but its legislative power is inherent in Parliament itself. Her Majesty has no independent legislative power, other than some remnants of the royal prerogative, and those remnants are always subject to being change or abolished by the Parliament.
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Parliament does not adminster. That is the task of the executive, which is accountable to Parliament and must carry out its functions in accordance with laws passed by Parliament, but Parliament does not adminster government programs.
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Parliament no longer has any judicial functions, following recent constitutional reforms which resulted in the House of Lords losing its role as the ultimate appellate body for most matters of domestic British law. Even before that change, the House of Lords did not exercise judicial functions as a delegate of Her Majesty, but as part of the inherent powers of Parliament - the “High Court of Parliament”.
The Constitutional Reform Act 2005 which created the Supreme Court and ended the Judicial Committee of the House of Lords did not specifically abolish the inherent power of Parliament. There has been some academic opinion that an appeal by way of Petition could lie from a judgement of the Supreme Court to Parliament, although it is something that would never happen.
Interesting - thanks for the clarification!