The website/sefer Halachically Speaking Vol.3 here says:
The poskim say one should not cut his nails or hair even when Rosh Chodesh falls out on Friday and wants to do so for kovod (for the honor of) Shabbos1.
Although some say one can be lenient2, the custom is like the first opinion quoted3, and one should cut them on Thursday instead4.
(Earlier he brought in the footnotes: Some say if one normally cuts his nails every Friday you can do so even on Friday Rosh Chodesh (Maharam Brisk 2:99, Orchos Chaim (Spinka) 260, see Shivim Temarim 56-57).
Refer to Magen Avraham 260, Shulchan Aruch Harav 260:1, Moreh B’etzbah 136:page 42, Yesod V’shoresh Hu’avodah 8:1:page 382, Chasam Sofer 158, Kitzur Shulchan Aruch 72:14,
Lekutei Maharich 2:page 5 (old), Shulchan Hatohar 260:4, Darchei Chaim V’sholom 353, Aruch Ha’shulchan 6, Shemiras Shabbos K’hilchoso 42:49, Rivevos Ephraim 4:97:37, Shraga Hameir 8:74, Divrei Shalom 4:57, Natei Gavriel (Pesach 3) page 221, Chai Ha’Levi 5:47, Yisroel V’hazemanim 1:22:page 319.
Knesses Hagedolah 260:page 131, Yosef Ometz 37:4, Tehilla L’Dovid 260:1, Lekutei Maharich seder erev Shabbos 2:page 306 (new), Aruch Ha’shulchan 260:6, Shemiras Shabbos K’hilchoso 42:footnote 174 in depth, Rivevos Ephraim 4:97:35, 37. The Nishmas Shabbos 1:160 says if one forgot to cut them on Thursday he may cut them on Friday (meaning Thursday night) during
Horav Yisroel Belsky Shlita, see Chut Shuni Shabbos 4:page 319 quoting this as the opinion of Horav Korelitz Shlita, Doleh U’mashka page 173.
Shivim Temarim ibid, Darchei Chaim V’sholom 353, Kaf Ha’chaim 260:12, Taamei Haminhagim page 122, Nishmas Shabbos 1:159, Divrei Shalom 3:65. Some have the custom to
cut the hair and nails in this situation on Wednesday (Chai Ha’Levi 5:47).