Ideally the exhaust of stochiometric liquid oxygen + liquid hydrogen rocket would be fast moving chunks of ice at zero kelvin. In this way all the potential chemical energy in the fuel/oxidizer is converted into linear momentum with nothing wasted as heat.
I assume the combustion chamber's job is to completely burn the hydrogen and oxygen together to produce very hot molecular water vapor with a minimum of ionization. The vacuum nozzle's job is then to redirect the molecules into a unidirectional stream. This means removing all the random motion and rotational and vibrational modes and redirecting the energy into linear motion. This probably requires an infinitely large nozzle bell that perfectly reflects any impinging molecules (and thus doesn't itself get hot).
Real rocket engines are presumably far from ideal, so I assume the exhaust is quite hot - though not hot enough to radiate at optical wavelengths. So my question is: assuming a thermometer co-moving with the exhaust gas, what temperature would it register?