Does the Immigration, Asylum and Nationality Act 2006 apply to the employment of all non-UK citizens?
No. The Immigration, Asylum and Nationality Act 2006 does not apply to the employment of Commonwealth citizens with the right of abode in the UK and citizens of any country in the European Economic Area and Switzerland. From 1 January 2020, following the end of the transition period, which is now in place after the UK's exit from the EU on 31 January 2020, EEA citizens and Swiss citizens will be subject to the same immigration rules as non-EEA/non-Swiss citizens.
However, in order to avoid race discrimination employers should avoid making assumptions about job applicants' nationality or right to work in the UK. As a matter of course they should require all job applicants, including UK, EEA and Swiss citizens, to provide documentary evidence of their right to work in the UK.