I saw the following image while searching for this answer in Sung Hwan Kim's thesis Germanium-Source Tunnel Field Effect Transistors for Ultra-Low Power Digital Logic It plots power density in Watts per square centimeter. Data points are for microprocessors, but it also includes indicators for 'Hot Plate", "Nuclear Reactor", "Rocket Nozzle", and "Sun's Surface".
The value plotted for the Rocket Nozzle seems to be 1,000 W/cm^2. Is that a sort-of canonical number with many engines used for launch purposes being somewhat similar, or is that an extreme example?
The Sun
The solar constant is the total electromagnetic radiation power per unit area at 1AU (about 150 million km) and is about 1361 W/m^2. Scale that by $1/r^2$ $to the radius of the Sun (about 0.696 million km) and that's 6300 W/m^2 which agrees nicely with the plot.
A Rocket Nozzle
I'll work through one example as a proposed way to estimate this.
Merlin engine with the smaller nozzle for first-stage atmospheric operation.
From this answer and an image in this answer I'll call the exit diameter $D$ of 90 centimeters and so radius $R$ of 45 centimeters.
From Wikipedia's Merlin (rocket engine family) I'll use the sea level $I_{SP}$ of 282 seconds and thrust (force) $F$ of 845 kN to get the total mass flow rate.
$$\dot(m) = \frac{F}{v=gI_{SP}} \approx 305 \ \text{kg/s}$$
Start by assuming correct stoichiometry as an approximation CH2 + 1.5O2 → CO2 + H2O
I get that 23% of the mass flow rate comes from the CH2
or kerosene.
$$\dot{m_K} \approx 69 \ \text{kg/s}$$
The energy density of kerosene $U$ is about 43 MJ/kg. Add in a fudge factor of 0.8 for incomplete burning, and I get:
$$I = \frac{P=\dot{m}U}{A=\pi R^2} \approx \ 466,000 \ \text{W/cm^2}$$
or 466 times larger than the number in the plot. This means I'm dramatically mis-interpreting something about the plot or I've mad a mistake in my math.
So the exhaust is certainly cooled by the expansion and not all of the chemical energy released in combustion is still present as heat at the exit of the exhaust, hopefully most of it has been converted to directed kinetic energy.