I was in a canoe race on a cold spring day in cold water. At the beginning of the race I ended up jumping into the river in waste deep water. My jeans stayed wet the whole race. My legs were cold wet and immobile for over an hour while my upper body was 'working out' paddling the canoe.
All was good until the race ended. I got out of the canoe to walk the 1/2 mile back home. As soon as I started walking I felt the cold blood from my legs rush up to my torso & started seriously shivering. Continuing to walk made it worse. Made it home anyway & collapsed under a hot shower. I felt at the time that if I'd have tried to walk much further I'd have had a real bad day.
Unlikely. there will be very little of that cold blood to mix. Very little. Adding heat will cause your core temperature rise. What is more likely to happen is that the sudden vasodilation drops your blood pressure, and your heart, possibly still in it's stunned cold state, can't compensate with stroke rate or variability. Then you risk cerebral hypoperfusion in the face of increasing metabolic demand (warm cells run faster, burn more glucose, need more oxygen to burn that glucose)