Where do you want to go with the baby? If you want to leave the UK, your status isn't particularly relevant and will eventually change.
Commented Aug 1, 2021 at 7:50Some guidance here citizensadvice.org.uk/immigration/visas-after-a-divorce and gov.uk/visas-when-you-separate-or-divorce What citizenship does your child have at the moment? You’ll need your partner’s permission to take your baby out of the UK gov.uk/permission-take-child-abroad This seems like a complicated situation, if you’ve not done so up to now you should seek advice from an immigration lawyer IMHO
Commented Aug 1, 2021 at 7:56If your partner had already lived in the UK for at least 5 years before the baby was born then the baby is a British citizen (which doesn't prevent the baby from also being an Italian citizen). Otherwise, the baby will be eligible to register for British citizenship as soon as one of you receives settled status (not pre-settled status). The baby is possibly already an Italian citizen, and, depending on the nationality laws of your country, the baby may already have that nationality as well. @Traveller a lawyer with expertise in divorce and child custody would be indicated as well.
Commented Aug 1, 2021 at 19:24As far as your status is concerned, article 13 of the withdrawal agreement extends EU freedom of movement rights defined in directive 2004/38/EC to people in your situation:
Particularly relevant to you is article 13(2) of directive 2004/38/EC:
Without prejudice to the second subparagraph, divorce, annulment of marriage or termination of the registered partnership referred to in point 2(b) of Article 2 shall not entail loss of the right of residence of a Union citizen's family members who are not nationals of a Member State where:
(a) prior to initiation of the divorce or annulment proceedings or termination of the registered partnership referred to in point 2(b) of Article 2, the marriage or registered partnership has lasted at least three years, including one year in the host Member State; or
(b) by agreement between the spouses or the partners referred to in point 2(b) of Article 2 or by court order, the spouse or partner who is not a national of a Member State has custody of the Union citizen's children; or
(c) this is warranted by particularly difficult circumstances, such as having been a victim of domestic violence while the marriage or registered partnership was subsisting; or
(d) by agreement between the spouses or partners referred to in point 2(b) of Article 2 or by court order, the spouse or partner who is not a national of a Member State has the right of access to a minor child, provided that the court has ruled that such access must be in the host Member State, and for as long as is required.
Before acquiring the right of permanent residence, the right of residence of the persons concerned shall remain subject to the requirement that they are able to show that they are workers or self-employed persons or that they have sufficient resources for themselves and their family members not to become a burden on the social assistance system of the host Member State during their period of residence and have comprehensive sickness insurance cover in the host Member State, or that they are members of the family, already constituted in the host Member State, of a person satisfying these requirements. 'Sufficient resources' shall be as defined in Article 8(4).
Such family members shall retain their right of residence exclusively on personal basis.
That's a lot to unpack and in a situation like yours I would strongly urge getting professional legal representation. There may be ways to secure your right to reside in the UK based on article 13(2)(a) (but your marriage hasn't yet lasted long enough) or on article 13(2)(b) (but only if your child is an Italian citizen and you do get custody). The last paragraph sets out additional requirements but if you continue to work (even part-time) those would be satisfied.
@Traveller also provided useful links on this situation, including this page from citizensadvice.org.uk which bluntly states:
You should go straight to a legal adviser if your partner is from the European Economic Area (EEA) and was living in the UK before 1 January 2021. There are special rules in this situation.