This year, it occurred to me that I had it entirely backwards.
Standing silent for the siren may have been introduced into Israel as a copy of procedures done in non-Jewish nations, but it is not chukkas ha-goy. It is simply a natural human expression of solemnity in the face of tragedy. And such a response to death goes back to the Torah itself. When Nadav and Avihu were killed by fire, it says vayidom Aharon, "Aharon was silent." Iyov's friends sat in silence with him for seven days - they didn't recite Tehillim. The Gemara (Berachos 6b) says that "the merit of attending a house of mourning lies in maintaining silence." Contemplating matters in our minds is something that is very much part of traditional Judaism.
What about the siren? A siren can be seen as simply a way of alerting everyone to this avodah, just like the Shabbos siren. Or, it can be seen as recalling the wars in which many lost their lives. It can even be seen as very similar to the shofar blast, another type of horn which sounds and to which in response we stand in silent contemplation.
Standing silent for the siren, then, echoes traditional Jewish practices. It is not something that is simply copying non-Jewish practices, like schlissel challah, pouring lead, and many other popular rituals in frum society.
But saying Tehillim, on the other hand - just how traditional is that? In my monograph What Can One Do For Someone Who Has Passed Away? I demonstrated that in classical Judaism, one gives charity for the dead and one prays (such as with the Yizkor prayer, which is recited at Yom HaZikaron events). For one's ancestors and teachers, one learns Torah and does good deeds as a credit to them. Saying Tehillim for strangers does not appear to have any basis in classical Judaism; the earliest sources to discuss such things explain that it does not actually accomplish anything for the deceased. I am fairly sure that it is a custom that goes back no more than two hundred years (but I am open to being corrected!).
So which is the traditional Jewish way of responding to such things, and which is the meaningless custom of recent origin? Like so many other topics that we have discussed here, it all depends on whether you define Jewish tradition as starting in Biblical times and carrying on through Chazal and the Rishonim, or whether you define it as starting about a hundred years ago. Similarly, it also depends on whether you follow the rationalist approach of the Rishonim or the more recent mystical approach.
(Of course, it is still infinitely better to say Tehillim during the siren than to ignore it entirely!)