Not that we have a legislative right to privacy in the United States, of course. (We got religion, press, habeas corpus, etc., but not privacy.) The right to privacy in the U.S. flows from Supreme Court decisions, the leading of which conservatives are trying to get overturned, namely, Roe v. Wade. (Wait a minute, I thought conservatives were supposed to be *against* government intrusion into private lives? Not when sex is involved, I guess.)
Anyway, I remember hearing a few weeks back about government agents coming into private homes in the U.K. to photograph people's property, to ensure that they were paying the proper amount of taxes on it. And the U.K. has a well-established history of public video surveillance, much more than in the U.S. So I suspect folks in the U.K. would be more used to this sort of thing than we would.
But the use of RFID to track people in theme parks is not unprecedented, even in the United States. Remember those magic pins at Walt Disney World, that would light up when you passed certain locations in the park? RFID tags. [Correction (3/30): A reader reminds me that RFID tags are passive, so they couldn't have been used here. If RFID were involved, it would have had a light flicking on on a building or something as the user walked by. A better example would be a ride that was programmed with several animation options, triggered by the RFID tags worn by users or ride units that passed by.] They could have just as easily been rigged to do what Alton Towers will be doing.