Re: What has immigration status got to do with criminal punishment?
No. Prior to 1962 there were few restrictions on Commonwealth immigration, her parents had it easy and they weren't even fleeing a war zone:
https://en.wikipedia.org/wiki/Commonwealth_Immigrants_Act_1962
"Before the Act was passed, citizens of Commonwealth countries had extensive rights to migrate to the UK."
Likewise the 1968 bill made it even harder, and every bill since has raised the bar. Her parents may have emmigrated totally legally in their day, but that immigration cause would now be illegal and they would be illegal immigrants under the new rules.
I think its more that if she did anything to make immigration easier, her opponents would cite her surname as proof she's soft on immigrants, so she treats them like shit to prove otherwise.