Yes they absolutely did.
I know this because I've been in several of them. However it was only in the most dangerous places - the criticality labs, where there was enough nuclear materials that, if mishandled, they could generate a fatal radiation dose in a matter of seconds. Most nuclear power stations would not have had a criticality lab and hence not an alarm of this kind. Only places like a research establishment (where I heard them) or a fuel rod assembly facility would have had them.
The sound was a short electronic bip every few seconds, not a 'ha' or anything else that could be mistaken for a human voice. I suppose different places may have used different sounds, but the ones I visited were all the same. They were called 'confidence tones'.
The sound played 24/7 to indicate that everything was OK, and especially that the alarm system was functioning normally. The sound stopping indicated a problem with the alarm. I'm not clear if the the noise stopping was treated as an alarm - or what level of alarm.
I was only ever a visitor, but I was told by the people who worked there regularly that you got used to it quickly and really noticed if it stopped.