Is is illegal for a cop to utter the words "stop filming" and do nothing more than that?
First, the legality of the order very much depends on context.
Glik v. Cunniffe
It is firmly established that the First Amendment’s aegis extends further than the text’s proscription on laws “abridging the freedom of speech, or of the press,” and encompasses a range of conduct related to the gathering and dissemination of information. As the Supreme Court has observed, “the First Amendment goes beyond protection of the press and the self-expression of individuals to prohibit government from limiting the stock of information from which members of the public may draw.” An important corollary to this interest in protecting the stock of public information is that “[t]here is an undoubted right to gather news ‘from any source by means within the law.’”
The filming of government officials engaged in their duties in a public place, including police officers performing their responsibilities, fits comfortably within these principles. Gathering information about government officials in a form that can readily be disseminated to others serves a cardinal First Amendment interest in protecting and promoting “the free discussion of governmental affairs.”
...This is particularly true of law enforcement officials, who are granted substantial discretion that may be misused to deprive individuals of their liberties. Ensuring the public’s right to gather information about their officials not only aids in the uncovering of abuses but also may have a salutary effect on the functioning of government more generally, see Press-Enter. Co. v. Superior Court, 478 U.S. 1, 8 (1986) (noting that “many governmental processes operate best under public scrutiny”).
In line with these principles, [our circuit, other circuits, and various district courts] have previously recognized that the videotaping of public officials is an exercise of First Amendment liberties….
To be sure, the right to film is not without limitations. It may be subject to reasonable time, place, and manner restrictions. We have no occasion to explore those limitations here, however. On the facts alleged in the complaint, Glik’s exercise of his First Amendment rights fell well within the bounds of the Constitution’s protections. Glik filmed the defendant police officers in the Boston Common, the oldest city park in the United States and the apotheosis of a public forum. In such traditional public spaces, the rights of the state to limit the exercise of First Amendment activity are “sharply circumscribed.” Moreover, … the complaint indicates that Glik “filmed [the officers] from a comfortable remove” and “neither spoke to nor molested them in any way” (except in directly responding to the officers when they addressed him). Such peaceful recording of an arrest in a public space that does not interfere with the police officers’ performance of their duties is not reasonably subject to limitation.
In our society, police officers are expected to endure significant burdens caused by citizens’ exercise of their First Amendment rights. Indeed, “[t]he freedom of individuals verbally to oppose or challenge police action without thereby risking arrest is one of the principal characteristics by which we distinguish a free nation from a police state.” The same restraint demanded of law enforcement officers in the face of “provocative and challenging” speech must be expected when they are merely the subject of videotaping that memorializes, without impairing, their work in public spaces.
The rest depends on how you want to define illegal. Disregarding those limited exceptions, it would be an order not supported by law but given under the color of law. It certainly would be illegal in that it violates a fairly significant federal statute -- US Constitution Amendment I. If you want to define illegal as a violation of a criminal code, it's probably not per se. The uniformed services favor the term unlawfulwhich is probably a better descriptor in this instance. It's still six of one and a half-dozen of the other.
It would be illegal for the officer to seize my camera, since her order to stop filming was unlawful.