First you must understand that a person has to make every possible effort to find water for wudu, and then he has to make every possible effort to find a way to do tayammum. The concession applies when obtaining water is impossible or unlikely - not when it is merely inconvenient.
There are several methods to obtain water in the situation that you describe:
You can purchase a water heater with a storage tank. These combine a heat source with insulation to prevent the water from freezing.
You can store some water in a place where it will not freeze. For example, you can store it inside your living space. Or at night you can heat some water and then store it in a vacuum flask (thermos). Or you can put it inside any other insulated container such inside a refrigerator or ice box.
You can collect some snow, put it in a pot, light a fire and melt the snow. If necessary, you can use a tool like a spade or pickaxe to collect the snow or to break any ice. Electricity, firewood or some other fuel is available in almost every place where humans are settled.
Who is unable to do both wudu and tayammum?
Contrary to your situation, there actually are valid circumstances when a person is incapable of performing both wudu and tayammum. For example, if he is imprisoned in a filthy place and being deprived of clean water and soil. Or if he is tied up and unable to move.
Prayer of a person who finds neither water nor a means of tayammum:
There is difference among the madhabs on what such a person should do. The following opinions exist:
- He should perform prayers without purification. Then he should repeat them whenever he is able to do find water or soil. This is the majority view (Hanafi, Shafi'i).
- He should perform prayers without purification. If he ever finds water \ soil, then repeating the prayer is not obligatory. (Hanbali)
- He does not need to perform these prayers. Purification is a condition for Salah, so when he is unable to do that the obligation has been waived from him. (Maliki)
Ref: Fiqh al-Islami wa Adillatuhu and Islamic Jurisprudence According To The Four Sunni Schools