i doubt it. polar explorers dont need to wear balaclava's over their faces and they dont die en masse from hypothermia.

the heat from the bath water is more than enough to keep the vital organs functioning well.

i imagine that the heat and steam from the bathwater may even create a thermal barrier protecting most of the face from the cold

The answer to this question depends greatly on the temperature of the hot bath, the ambient temperature of the bath room and whether it is well ventilated, and the tolerance of the person in the bath, including his body characteristics such as body mass index, and the surface area of his body.

The most probable answer is no. To initiate hyperthermia on a moderate temperature usually involves more than 10 minutes. This is sufficient for water to be cooled by 1 or 2 degrees. Sugnificant discomfort will have encouraged the person to come out of the bath, quickly lowering the body temperature well before the person passes out.  While cannot be directly compared, a sauna session can reach up to 80 degrees celcius, aided by perspiration, most healthy adult can endure up to 15 minutes.

Personally I have been into 40 degrees Celcius hot spring in Japan for 15 minutes, it gave a similar experience as being a bit drunk and dizziness ensues, but I did not appear to be very ill.

Absolutely!  Even snorkelling in shallow water in the tropics can cause a similar problem.  In fact, if the humidity is high enough, even hot air can be dangerous.  Saunas are tolerable only because the air is relatively dry.  If someone adds too much water to the stones it can rapidly become unbearable as the humidity rises, and you can no longer lose heat by sweating.

Many years ago in Abu Dhabi I was working indoors in a workshop where the ambient temperature was nearly 50C and the humidity was 100%.  Sweat did not evaporate at all, and I would almost certainly have died from hyerthermia if I hadn't had an air-conditioned office to cool off in occasionally!