There are arguments the Vienna Convention has created acquired rights, and the ECHR's right to a private and family life. There's also the ability to apply for permanent residence if you've been in the UK for more than 5 years.

This is all being predicated on a more specific settlement not being reached with the EU. Given the UK says it wants one that shouldn't be a problem and may well turn into the UK making a unilateral grant of residence even if the EU decides to play hardball.