EHIC is not insurance. It allows you to get insurance at the same rate as normal citizens of the country and cannot be used at private facilities, only state facilities. So if the French pay for treatment, so will you. Also there are strict rules about maternity, and if it looks as if you came specifically to give birth, they can require you to pay.
In order to travel back to the UK with the baby, you would have to register the birth at the British embassy and wait AGES for the passport. They aren't automatic French citizens just because they are born there. But before you can register the baby you need to get the birth certificate. I don't know how long it takes in France, but it was 8 weeks after the birth before we could get an appointment to register the baby to get the birth certificate. Then it's getting a appointment at the embassy, then waiting for documents. Your wife could be stuck there for months.
Would it be possible to bring the interview forward, or possibly haing them come to her given the situation??
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