Having read a few discussions regarding the Plantagenet's and ascending line of monarchy since, I became curious if another group / family had a 'more' legitimate claim to the thrown (asked and answered, though not sufficiently in my opinion) or if another family could become the monarch. I don't want to base a question on an assumption, but I am curious all the same if by whatever manner the Windsors could be replaced - for lack of a more appropriate term.
Note: I don't think that what I'm going to offer is a particularly good answer, but I think it is a worse comment.
I think the key to the answer is the Glorious Revolution in which England expelled James II and invited William and Mary to rule England.
Aside: @Oldcat's comments on the Young Pretender and the Old Pretender are very pertinent. I approach the question through the lens of an American fascinated with British Constitutional evolution in the period. I think that @Oldcat's comment is probably closer to the OP's actual intent. For me, the British Constitution prevents any pretender from having a legitimate claim on the throne.
Although the Glorious Revolution is vital to understanding Great Britain, in my opinion it isn't the actual event of the Revolution, but the events of the following years that really define the British Constitutional Government. I'd recommend starting with Edmund Burke Who is brilliant, but not easily quoted or summarized. With respect to the current question, I think that Reflections on the Revolution in France is vital, and I believe that Thoughts on the Causes of the Current Discontents may also be useful. I actually haven't looked at this question in a couple of years, but I believe that the (British) Bill of Rights of 1689 is probably important. Essentially Burke argues that the English invited William and Mary to rule only after James II had invalidated his rule, and that the English agreed to be bound to the rule of William and Mary. He is quite adamant that unlike the French, the English Constitution is not based on a social contract, and that the English don't have the privilege of selecting Monarchs willy nilly.
Updated aside: I haven't researched this period for about five years. I remember finding a paragraph from Burke that I thought summarized the question admirably (once I decoded Burke's prose). I'm truly frustrated that I cannot find that quote anymore (I've begun to wonder if it was not Burke, but someone else, but I cannot remember).
If you're looking for a good overview, I would suggest Mike Duncan's Revolutions podcast and website - he covers the English Civil war fairly well. Although the Civil War is distinct from the Glorious Revolution, I understand the Glorious Revolution better for understanding the Civil War.
I also recommend @Kobunite's answer summarizing the effect of the Bill of Right. I think Kobunite's summary of the various clauses is clear, helpful and accurate. I wish I had the skill and the knowledge to explain why it is difficult for a modern reader with a modern reader's understanding of constitutional theory to read the Bill of Right and understand the assertions. Modern readers rely on assumptions and concepts that were formed as a result of the Bill of Right; the contemporaries didn't have that advantage. Quite frankly the analytical summary that Kobunite provides is (a) why I find the period fascinating and (b) part of what makes this period so vital to the understanding of the American government.
Today the British monarchy is an almost entirely ceremonial position. All the real political power in the country is invested in Parliament (mostly the House of Commons) and the court system. The Queen is left with only the ceremonial "Head of State" duties that in the US are performed by the President (or more often in fact delegated to the First Lady and/or the Vice President). Greeting foreign dignitaries, attending state funerals, etc.
For this reason, frankly there's no real incentive for anyone to try to change things in the violent manner that would typically be implied in an "overthrow".
Parliament can however, and in fact often does, pass laws changing both the personal requirements for a monarch, and the succession requirements. This generally has to be coordinated with the other dominions to avoid inconsistencies. This was most recently done in 2011 to allow for a fully gender-neutral succession. It would similarly be in their power to depose an unacceptable monarch (as was effectively done in 1936).
The most likely scenario for any kind of radical change though (and probably still not very likely) would be an abolition of the monarchy altogether, with the Head of State position becoming an elected one (or perhaps selected by Parliament). This gets discussed a fair bit in England particularly whenever any personal problems of Royal Family members make news, but support for that idea recently seems to bump around between only 15%-25%.
The current monarchy in the United Kingdom is very attuned to the public relations between that monarchy and the people of England (yes, and I mean only England). Other public relations behaviour is far more muted in comparison. This responsiveness emerges from a number of public relations disasters, including the monarchy's German heritage which caused the formation of the House of Windsor itself (https://en.wikipedia.org/wiki/House_of_Windsor#Foundation).
This sensitivity to the public is probably due to 20th century incidences of dethronement in Europe after mass public unrest, generally leading to a parliamentary republic, occasionally and briefly leading to systems of workers councils. This, at least, is the animating threat that lies behind the the House of Windsor's activity, service to the nation, etc. I would suggest that fear from the Windsors indicates at least one viable method of removal, within or without the constitution: mass public outrage which either parliament attends to, or which also destroys parliament.