They will just do an emergency c section, and that doesn’t mean it’s an emergency they just class it as an emergency because it didn’t happen when it was planned too! I was booked in for 39 weeks due to health too, and was also worried about the same thing and this is what they told me, I’m now currently in hospital tho with preeclampsia so could possibly be brought forward anyway
I asked this question (mine’s on the 14th!) basically if your waters break you go straight in (let them know you’re coming) and you basically go in as a low grade emergency c section so they do it as soon as they can. She said in rare instances like if things are moving too quickly there might not be time (which is seriously causing me anxiety too!) but in most cases you still have the c section xx