The monarch of the UK in the 20th century played a huge role in government because they appointed everyone in the House of Lords. The monarch still has that role formally, but, today it is unlikely that the monarch will veto any person recommended by the Prime Minister. In 1997 many Lords were removed and the Prime Minister got the primary power of proposing new Lords for appointment.

The last time, by my count, that the monarch played a significant role in British government was 1911. At that time the elected House of Commons was proposing increased social spending that the hereditary and appointed House of Lords didn't want. The Lords vetoed the budget and they thought this would force the Commons to compromise. It didn't. The King took the side of the Commons and chose to appoint a large number of new, Liberal Lords who would outvote those who were already there. Had the King remained passive, the road to a democratic welfare state would have been more slow and perhaps more violent. However I can not think of any more recent incident where the King or Queen's opinion mattered on a question of government.

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