Happened twice with me. I'm not sure what your consultant meant by the c-section being dangerous if you're well established in labour?!
I tried for natural birth with my first, but after 10 hours he was stuck at 5cm dilation (he was too big). I had an emergency c-sect and baby was born healthily.
With my 2nd, I was supposed to either try for VBAC, or have a c-sect around 39 weeks if I didn't go into labour myself. Unfortunately, at 37 weeks the placenta failed spectacularly, and I had another emergency c-sect because baby's oxygen was cut off and she had to come out immediately. She spent some time in NICU before coming home.
With my 3rd, I was again supposed to have a scheduled c-sect at 38+4 weeks, but at 37+0 I went into spontaneous labour. I was afraid his lungs won't be ready either, so the dr was able to stop labour every day for 4 days. By then I went into full-blown active labour. It was around 10am in the morning. The c-sect was only done after 3pm the afternoon, I have no idea how far labour has progressed, but I'm pretty sure I've dilated strongly by then (contractions was 2 minutes apart). Still, the c-sect was done without any problems and baby was born with healthy lungs.
I'm scheduled again to have a c-sect around 38+4 this time, but both me and my dr know I won't reach it. So once I go into labour, I'll just proceed to the hospital where the c-sect will be done.