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I am afraid that the Zambrano route would only be open to you if you were in an ongoing relationship with the biological father of your baby.
You will still be entitled to child maintenance even if your ex doe snot let you put his name on the birth certificate but it will not enable you to stay in the Uk - and indeed even if you were still in a relationship it is not guaranteed - Zambrano would only work if the child had already lived here for a number of years
Please ask if you need further details
If he is the biological father you can either apply for child maintenance via the CMS if you are in the Uk or through the courts if you are not .
You do not have to live in the UK to sue him for the Child Maintenance.
He will have to pay 12% of his gross income.
You can also apply to the court for a Declaration of Parentage to prove that your child is entitled to British Citizenship - but this will NOT mean that you can stay in the Uk automatically.
If your new partner marries you then you can go home to Cameroon and apply for a Spouse visa.
It does not matter that you have no proof.
If you apply for maintenance whether to the CMS or to the Court then HE has to prove that he is NOT the father - not the other way around.
If he has a DNA test done then you can use that to obtain a Declaration of Parentage.
It is always possible to trace using an Enquiry Agent of he moves away.
So long as he is in the Uk then you will be able to take these steps
It is the law that you can only change your visa status to a spouse visa from OUTSIDE the UK - although in fairness it can be from anywhere outside - you need not return to Cameroon to do it
With regard to Zambrano the issue is a Right to remain for the parent of any EU citizen if there is no other way that child can exercise its right to remain in the EU.
The UK has in fact said that there is no reason such a child should not live with the father in such a situation - and it is also likely to be amongst the first casualties of Brexit.
There are no other benefits I am afraid.
The father could claim Child Benefit - but you cannot because of your immigration status.
It will take time to obtain the evidence that would make Zambrano relevant.
It is then likely that in fact all such decision will be delayed whilst the negotiations are ongoing.
Indeed it is arguable that once Clause 51 has been triggered then the Rules will not be applicable in any event. Sadly immigration is an area where I fear they will act sooner rather than later
Even if you were able to I remain here using Zambrano you would not be entitled to accommodation other than basic temporary accommodation nor to benefits at the level that others receive them.
My concern is that there is Case law which has stated that a mother in similar circumstances has no right to stay - although that decision is being appealed.