According to the Gov.UK website, one must give a week's notice if they wish to resign from their job and have been working for at least a month. (Source)
Why does this one-week rule not apply to the Cabinet?
The notice period is set out by the Employment Rights Act 1996, and is specific to employees. Ministers are not working under a contract of employment, or, for that matter, any other sort of contract. That means they are not "employees" nor even "workers" under UK employment law. The most appropriate status is "office holder", for which the idea of "notice" (or for that matter "unfair dismissal") generally doesn't apply. The same is true for all MPs.