What is the ruling on the prayer of someone who finds himself outside the home when the time for prayer is due, and is prevented from performing ablution (Wudu') because mosque restrooms are closed? Is it permissible for him to pray without ablution, or to perform Tayammum (dry ablution), or to perform ablution from public water taps that are sometimes available in the street?

Ablution (Wudu') is a condition for the validity of prayer for one who is able to perform it. If you can perform ablution from street faucets or by purchasing water, then that becomes obligatory, and dry ablution (Tayammum) is not sufficient for you. However, if you cannot find water, even by purchasing it, and you fear that the prayer's time will expire, then you perform Tayammum.