Is a mother obliged to cut ties with her adult daughter who has declared her abandonment of Islam, in obedience to her husband's command?

If your Muslim daughter apostatizes from her religion, she must be boycotted and shown animosity until she repents, especially if the husband orders it, due to the absence of a system that enforces the apostasy penalty. This is what Ibn Muflih (may Allah have mercy on him) said, citing the words of Imam Ahmad and Ibn Aqil.

However, if the daughter was not originally Muslim and her parents embraced Islam, there is no harm in communicating with her, speaking with her, and trying to guide her to Islam, as Umar ibn al-Khattab (may Allah be pleased with him) did when he gave a gift to his polytheist brother. Boycotting differs between an original disbeliever and an apostate; the former may be reconnected with for the purpose of دعوت (invitation to Islam for guidance), while the latter is boycotted so that he may be deterred.