davidm, there is nothing wrong with the employer and employee agreeing a net pay arrangement. For example, the arrangement being that £x net per month is paid. However, in the background, someone needs to do the necessary calculations and calculate the gross equivalent of the net pay, and then pay over the correct amounts to the Inland Revenue.
As regards NI, you say "since she is an expat employee, her employer is not making any NI contributions.". OK there are special rules for NI for those arriving in the UK, dependent upon whether the person is from the EEA, from a country with a reciprocal arrangement with the UK, or from any other country. Quite a complex matter. Much more information provided at :-
Special Cases:Contents
-: or rather at the links off that page.
But your wife has been working in the UK? For how long? Or outside the UK? Or indeed both?