Look, I'm sorry, but this is incorrect.
Please don't scaremonger. This is stressful enough for the OP without leading her to believe that this could result in there being a warrant out for her arrest.
The court summons - if it is a real court summons which it may well not be - is invalid for two reasons.
- It was not served to the correct address.
- The court has no jurisdiction in this case.
A court summons must be served to the correct address in order to be valid. Admittedly if there is no response the court has no way of knowing whether it was not correctly served or the recipient simply ignored it. It is possible that if she does not respond the court may issue a default judgment in favour of the father. However, if this happens, and the OP later becomes aware of it, all she has to say is that (1) the court summons was invalid because it was not sent to the correct address; and (2) the child has been habitually resident in the UK since birth and so the French court has no jurisdiction. Either one of these facts would be enough to have the default judgment set aside.
It is possible, but unlikely, that the French court might issue a default judgment in favour of the father and then ask the UK court system to enforce it. After investigating, the UK court system would then decline to enforce it on the grounds that (1) the court summons was not correctly served, and (2) the child is habitually resident in the UK and so the French court has no jurisdiction. And that would be the end of the matter so far as the French court is concerned. They may then order the father to pay legal costs, but that is not the OP's problem.
There will be no warrant out for the OP's arrest and no issues at any EU border, because the European Arrest Warrant only concerns serious criminal conduct and not minor civil law issues such as this. As such, the OP would only be at risk if the father alleged that she had kidnapped the child, in which case there would be a police investigation in France before any arrest warrant could be issued. And even the most cursory police investigation would soon reveal that the child was not born in France, is not known to the French authorities, has not yet been recognised as a French citizen, there is no legally recognised filial link between this man and the child, there is no record of the child ever having lived with this man or ever having lived in France, and so there is no evidence whatsoever that the OP could possibly have kidnapped her child from France.
Basically this man is being very silly, trying to scare the OP into dropping her claim for child support. Pursuing this matter will most likely cost him an awful lot of money in irrecoverable legal fees and possible court fines, and it will not absolve him of his obligation to pay child support or get him any access to his child (assuming he even wants that). Anything he does in France is a total waste of time and (his own) money because France has no jurisdiction here.
If he actually wants contact with his child he will need to first apply to the UK court to be added to the birth certificate, and then start proceedings in the UK regarding the child's residence arrangements.
He has absolutely no hope of getting any kind of shared residence unless he gets a visa to come and live in the UK, finds a job and a place to live which is close to where the OP lives and where her child is at school, and applies to the court then. The UK court is certainly not going to order that the child goes to live in France with a near stranger.