So, just to save me reading the whole thread. If we rescue migrants in French waters, which bit of law says France has to accept them back.
If a UK vessel rescues those in distress in French waters, the whole rescue will probably be coordinated by the French.
In which case they may be content for a UK vessel to land them in France. Strictly speaking, they can't refuse.
But if the French are coordinating a rescue and tell the UK vessels that their assistance is not required, and the UK vessel still rescues some refugees, the French may well refuse to accept them.
But the UK vessel would go to the nearest port only if there was a medical emergenccy that required it.
If the UK vessel has some rescued refugees who need urgent medical attention, and don't have the facilities on-board, the French would allow them to be landed if that was the best option.
There isn't a competition nor a conflict between the French SNSM (Societe Nationale de Sauvetage en Mer) and the UK RNLI
They frequently act together in rescues, and they both have the same goals, to provide a rescue service on the water.
The French SNSM is a voluntary service as is the RNLI and cannot be controlled by the government.