Is it permissible to boycott one's female cousin who has committed the major sin of fornication and given birth to a child, and is persistent in this disobedience despite being reminded, and does this boycott affect the acceptance of one's deeds?

Your cousin committed two grave sins: abandoning prayer and fornication. She must be advised and made aware of the enormity of her situation, and called to sincere repentance, especially for abandoning prayer. Perhaps her prayer will deter her from indecency and evil. Your uncle should be firm with her. If she repents, then praise be to Allah. If she persists in her deviation, she should be forsaken if there is hope that forsaking her will be beneficial; otherwise, maintaining ties with her is more appropriate. What was mentioned about her family's lack of reaction is considered dayathah (moral laxity/cuckoldry).