Suppose a man falls into very cold water and gets their foot stuck under a heavy rock. Fortunately, his head is above water and someone is able to call for help. The paramedics want to keep him warm while they work on freeing his foot. They put a hat on his head. Should they also wrap him in a blanket?
There are three mechanisms of heat loss (this applies generally, not just to the man in this example) radiation, conduction and convection. In most everyday cases radiation can be neglected so we just have conduction and convection.
Conduction is just the transfer of heat along a static object. For example if you hold the end of a metal bar in a flame pretty soon the heat will conduct along the bar and the end you're holding will get hot too. In this case if our man is surrounded by a region of still water he will lose heat by conduction into the water.
Convection is the transport of heat by a moving fluid. For example if you stand in still air on a winter day it may not feel too cold, but if a gale is blowing you'll start feeling cold very quickly. This happens because your body heats the air immediately around it then the wind whisks that warmer air away and replaces it with cold air.
Now back to your question. We can't do anything about conduction. The man is in the water and we can't change the thermal conductivity of the water. But we can do something about convection. Our man's body heat will soon heat the water immediately around him, and we don't want water currents carrying away that warm water and replacing it with cold water. Wrapping a blanket tightly around the man will trap a layer of water near his skin and prevent convection from cooling him.
The blanket would help a bit in the same way it keeps the body warmer in cold air, by inhibiting convection and allowing an interface of warmer air between the blancet and the body, but water is a better heat conductor than air and it will not be very efficient.
Divers who stay long in the water have water suits whose material is designed to keep the body temperature comfortable even in quite cold water. Different types for different waters.
In the emergency you are talking about cutting up a wet suit will offer better protection.
p.s. when I was young we had woolen bathing suits, and it is true that they are warmer than the synthetics we get now.
Blankets do not "warm you." Blankets slow down heat transfer by convection, by simply restricting the movement of a medium which may carry heat. Air absorbs heat from solid objects by conduction. Air does not conduct that heat to other air molecules very well. So, if you restrict the movement of those air molecules, you will limit the ability of that air to carry that heat away from the solid object from which it originally absorbed the heat. As a secondary factor: The closer the temperature of the air is to the temperature of said solid object, the less heat absorbed by the air. So, as that trapped air absorbs more heat and rises in temperature, the less heat it can then absorb.
The Blanket will help but the amount depends on the type of blanket cotton or other fabric blankets will be of fairly small help because the water can still flow through them just not as quickly as it would without the blanket in the way where as non porous material blankets would be of significant more help as they stop the flow of the water carrying away more heat
I suspect that the only significant benefit that you can achieve from a thin layer of cotton is obtained wrapping it tightly especially around the limbs so that the blood is forced to stay deeper dissipating less heat.
Although it is for sure true that a thin layer of cotton will somehow limit the convection, to effectively contain the heat losses you need to create a thick layer of static water and looking at the thickness of diving gears, I believe that the effect of a blanket is almost negligible if just loosely wrapped around the unlucky man.
It definitely will and I can attest to this. During my swim career while waiting on the wall for your set there was always some discomfort when kicking of the wall after having been on it for a moment or two. The discomfort is because a "shell" of water around your body, about an inch, had been warmed up and remained there assuming no one had swam by you and the water was calm. You could feel this on other people as well. So a blanket around you would buffer against natural currents in the water and if it's thick enough, doesn't convect heat well, and the temperature difference isn't to large it will keep you warm. This is how a blanket works when dry by the way. It simply keeps the air inside warm. The specific heat of air is about 50% greater than the specific heat of cotton according to engineeringtoolbox.com. It is the still air inside that does the insulating, not the fabric.
Simple answer is no!! Blankets only work in air. Since air is a bad conductor of heat and blacnkets use this air to provide as a cushion for your heat and the external environment. Since in water there wont be much air and water too is a bad conductor of heat. Hence you should not use blanket in water
Short answer, no, it will not warm the body underwater.
Consider, what is the temperature of cold water? If cold water is that of the arctic area or thereabouts, which is usually about 5 to 10 Celsius, then those paramedics better work fast, blanket or not. Be that as it may, remember to calm this man down first by reassurance/s as panic just increases the stress on the body, and that water, whether trapped or not in neoprene or woolen blanket, will rob the body heat.
Water below 21 C (~70 F) is generally considered cold, and that of Pacific Ocean even in San Diego/CA area will be a chilly 10 to 15 C (~ 50 to 60 F). If I was the rescue team [paramedic], I would highly recommend putting the face into that very cold water as this would invoke the mammalian diving reflex pathway. It may seem inhumane, but will increase the chances of recovery post-rescue. Then work on unsticking of the foot. Have a read from here, http://bit.ly/TWf3f8 for cold water survival/rescue.