The UK doesn't have a written constitution, but it does nonetheless have a means of administration prescribed by parliament - a constitution.
Some laws are held to be 'constitutional'. The Bill of Rights 1688 for example (on which the American Constitution was partially based if I recall correctly), which limits the power of the monarchy. The Human Rights Act 1998 is another example.
It is of course possible for parliament to abolish any one of these laws - and I believe the same is the case in America, although a special majority of some sort is required to ammend the constitution. Beyond that somewhat technical difference, it's a similar system.
By convention, the Queen is said to have the right to be consulted, the right to advise, and the right to warn. Technically she has a right to veto legislation, but this last occurred in 1709 and if she attempted it today, I suspect there would be a constitutional crisis leading either to a general election or a referendum on the monarchy.
As it happens, the current monarch is noted for having been reasonably good at her job. She has experience of eleven Prime Ministers and the events of half a century - whether or not you agree with her constitutional position, she is currently an extremely useful resource to the government.