What does 'after commencement' mean in the following sentence?

Commence means start.

So it means the clause will apply after the act comes into force.

As clarified in the marginal notes in the text of the act itself,

"[C]ommencement", without more, means the commencement of this Act.

In other words, commencement refers to the moment law is enacted.

Commencement for each part of the Act was the day set out in an order under subsection 53(2) which says

This Act, except the provisions mentioned in subsection (3), shall come into force on such day as the Secretary of State may by order made by statutory instrument appoint; and references to the commencement of this Act shall be construed as references to the beginning of that day

Only for sections 49 and 53 was commencement the day the Act was enacted. According to the UK Border Agency, the BNA 1981 received Royal Assent on 30 October 1981. Its main provisions came into effect on 1 January 1983.