Given the British citizenship and UK residency rules that we have (why make something easy when you can make it pointlessly complicated?), it seems we'd need to hold information back up to grandparents. So those Windrush landing cards needed to be held for maybe 120-150 years.
Alternatively, the Home Office could get with the times come up with a system that means you don't need to keep years of paperwork up to your grandparents. Everyone would ask for residency on arrival, everyone would register with the council every time they move house, and everyone would be able to prove continuous residency five years later by granting the Home Office permission to look at residency/tax/NI histrory. Same for citizenship five years after that.
It seems to me they were aiming to go in that direction, but 1) didn't want to regularise because that would "send the wrong signal" but is essential when switching from the system we have now to a system like this, and 2) did it on the cheap by using the public as immigration officers, instead of just asking them to say "show me this card or certificate, if you haven't got one, get one from this place" right at the start, and 3) didn't give a toss about how many people's lives they screwed up as we can see by the fact that their hostile environment catches everyone who came before 1973 if their country wasn't independent.
Also Mayhem will probably use the opportunity to push for ID cards for everyone saying it can't be done without it, but countries like the Netherlands manage to operate a more coherent immigration system and have optional ID cards.