People might complain, if they knew. It's happened. Such complaints have even led to expensive litigation and ultimately informed the present UK federalism arrangement.
tl;dr it's not as simple as calling herself what she wants whenever she wants, but also nobody in any government would refuse a reasonable plan to make a change of style.
I'll restrict myself here to UK law. Laws of other countries apply to her use of styles in contexts relating to them (such as when she is physically visiting them), as well.
Formally, styles of address are the personal prerogative of the reigning monarch, and she has been delegated by Parliament the full right to style herself as she sees fit under the Royal Titles Act (1953). Exercising the prerogative _lawfully_, however, would require her to at least inform the government and have the matter Gazetted.
The statute (http://www.legislation.gov.uk/ukpga/Eliz2/1-2/9) clearly gave her the right to style herself reasonably, and that statute is probably not exhausted (there is a plausible argument that the delegated authority ceased to exist upon first use, and another that the agreement has expired because half the governments no longer share the same monarch and New Zealand introduced its own _conflicting_ statutes including the Royal Titles Act (1974, New Zealand)).
However, even if the statute is fully valid, the government-of-the-day can technically insist that she not use the delegated prerogative without formal advice from a minister. It is likely that both the department of constitutional affairs at the Ministry of Justice (which has among other things formal responsibility for policy relating to the Royal Household, the personal representatives of the monarch, and "non-delegated" royal and ecclesiastical prerogatives) and the Foreign Office (since there is still a personal union of the crowns of several independent countries) would push hard to exercise the right to give formal advice. On the other hand, most governments would simply rubber-stamp any reasonable change quickly, or even retroactively, and arrange its publication. That's important since judicial scrutiny of personal prerogatives can (and sometimes does) happen and a "slip up" by the monarch could in principle expose the government to substantial liability.
(A concrete example is in MacCormick v Lord Advocate 1953 SC 396, 1953 SLT 255 which was a case decided under Scottish Law involving the current monarch's choice to style herself Elizabeth II. The courts found, essentially, that the monarch had acted lawfully in styling herself Elizabeth II, and the decision refers to minutes and letters by government ministers endorsing her name(s), that the supporting statute was valid with respect to MacCormick & Hamilton's claim, and that her regnal name was widely published, and in particular was Gazetted here https://www.thegazette.co.uk/L... -- note that "Published by Authority" means by the authority of the government, not the monarch, and "with the advice of our privy council" means that there was a formal consultation with the government.
MacCormick was a Scottish Nationalist lawyer and Rector of the University of Glasgow, and raised the matter as a point of interesting differences in Scottish and English constitutional and administrative law; Hamilton is a well known Scottish Nationalist lawyer; both were trying to establish that Scotland had the right to be consulted on matters of personal prerogative. In effect, although they lost this particular case, their ideas are now entrenched in devolution law -- a change to the monarch's style would almost certainly have to be supported by an Act of the Scottish Parliament. In particular, MacCormick was specifically referenced in the parliamentary debates leading to paragraphs (1) and (2) of Part I of Schedule 5 of the Scotland Act (1988) -- it is likely that in litigation in the Supreme Court of the United Kingdom, the monarch's style (in Scotland) would be found to be an unreserved matter under (2)(a) rather than covered by (1)(a) or implied in (3)(3)(a).)
So in short, Unelected Ceremonial President Windsor has to abide by various Settlements which require her to give up many of her personal freedoms (speech and expression, religion, association, full free choice in marriage) in exchange for continuing use of vast tracts of land, palaces, money, jewels, publicly funded servants, and so forth. That's been the case in most constitutional and even many absolute monarchies in Western Europe since no later than the 15th century.