May we say tehillim on nittul starting from after chatzos by day?
One may say tehillim the day before nittul, as the minhag only applies at night until chatzos, even for thse who abstain from learning on this night. Secondly there are different minhagim regarding not learning on this night, and different nights that it might apply, (Dec. 25 or Jan. 6). The Sefardim and the litvisher yeshivos do not change anything because of nittul, however the Chasidisher minhag is to be careful about this. If one has the minhag not to learn, it would include not saying tehillim from at night until chatzos.
As a side point, I am attaching a link to an article published from this site regarding this topic. I think you will find it helpful and enlightening. It discusses, the history of nittul, when it applies, the reasons for it, and who has this minhag. https://dinonline.org/2011/12/20/nitel-nacht/
May Hashem send us Moshiach tzidkeinu, and obliviate all of the avoda zara, b’mehira b’yameinu, so that such questions will no longer be applicable.