If I live in a house that my mother owns and has a mortgage on but she is ill, how I can get on the title or have the house gifted to me?

Title to the house is different from the mortgage.  Title depends on the deed.  A mortgage is a separate document.  It is possible for your mother to transfer the house to you by deed (either Quit Claim Deed or Warranty Deed).  Your mother could also transfer title to the house from her name alone to her name and your name as joint tenants with a right of survivorship.  That way, you would inherit the house if your mother passed away.

Changing title to the house, or changing the deed, may have consequences with the mortgage company.  Some mortgages say that the company has the right to demand the entire balance if the mortgage-holder transfers title.  Whether the company will do this, or whether they will even notice that title has been transferred, is uncertain.  If you or your mother keep paying the mortgage, they may do nothing.

Remember that a mortgage is secured by the property.  If no one pays the mortgage, the company can take the property in foreclosure, no matter who holds title on the deed.