I'm Dr Chaithanya, experienced General Physician attending you.
The question has been just posted to doctors.
Don't worry. Bell's palsy happens in rare cases when exposed to severe cold for a prolonged duration.
As you said, your exposure is limited to 30min and only your head is exposed to cold, nil chances of you getting into the similar situation.
Now, you did a right thing by taking a hot shower. Keep yourselves warm, and avoid stress.
Stress and thinking much about it can do more harm than it has to happen, so be assured, nothing is going to happen in your case.
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