"Cut no ice"comes from the US,but the origin of this saying is not clear.
In daily life,we say:
*His wealth cuts no ice with me.I'll never marry him anyway.
*Paul's advice cuts no ice with his son.
Have no effect, make no impression, as in That excuse cuts no ice with me. This term predates modern refrigeration, when ice was obtained by cutting it from a large block with a sharp tool. [Late 1800s]