When a person is alone in the bathroom is it permissible for him to do Wuduu while he is totally naked?
Yes, it is permissible for him to do so. Being clothed is not a condition for a valid Wuduu. However, it is best that one is covered. This is because it is not fitting for a person to be naked, even when alone, unless it is for a need.
This is based on the following hadith which has been graded hasan by certain scholars
Bahz bin Hakiim said that his father told on the authority of his grandfather:
I said: O Messenger of Allah, from whom should we conceal our private parts and to whom can we show? He replied: conceal your private parts except from your wife and from whom your right hands possess (slave-girls).
I then asked: O Messenger of Allah, (what should we do), if the people are assembled together?
He replied: If it is within your power that no one looks at it, then no one should look at it.
I then asked: O Messenger of Allah if one of us is alone, (what should he do)?
He replied: Allah is more entitled than people that bashfulness should be shown to him.
[Sunan Abu Dawud Hadith No. 4017]