Legally speaking, very little - Thompson would clearly be acting within the law, so the whole thing would rapidly be brushed under the carpet. Politically speaking is more interesting - it removes de Portes at time of huge pressure on the French government and when her influence can only be described as malign. Her death will leave Reynaud in even more of a mess than he already was in, but in the grand scheme of things that will have little influence - he was badly shocked by the whole situation and barely holding things together as it was.
So at worse absolutely no effect, at best it pushes towards some sort of France Fights On scenario - possibly with a Petain regime in Metropolitan France and a Government in Exile reaching beyond just De Gaulle in Algeria, protected by the RN (and with most of the MN loyal to it).