What would the bold line mean?
"That day" refers to Saturday. The British lawmakers don't usually work on Saturdays, and the last time they did so was in 1982.
"have been" is present perfect. Why is that tense used?
There are many different situations where the present perfect tense can be used. It can be used in the following ways:
To describe an action that is being repeated between the past and present.
Example: We have gone to the beach many times.
To describe an action that started in the past and is still continuing in the future.
Example: I have lived in the United States since 1990.
Even if "next Saturday" is in the future, all of the other Saturdays since 1982 are in the past. Saturday is, roughly speaking, in the present era when compared to 1982.
So, present perfect "have been" seems to fit.