Praise be to Allah.
There is nothing wrong with wearing fake nails permanently if that is because the natural nails break due to a lack of calcium in the body.
But if they are worn for the purpose of adornment and beautification, that is not permissible. Please see the answer to question no. 21724 .
There is nothing wrong with praying whilst wearing these nails, but it is stipulated that they should be removed when doing wudu’ and ghusl, so that the water may reach what is beneath them.
It says in a fatwa of the Permanent Committee (5/218): If the nail polish forms a layer on the surface of the nails, then wudu’ is not valid unless it is removed before doing wudu’. If it does not form a layer, then wudu’ is valid, as in the case of henna.
If this is the case with regard to nail polish, then it applies even more so to artificial nails.
And Allah knows best.