It would be more accurate to say there is no single register of addresses.

The electoral register knows I’ve left the U.K.; I had to tell HMRC separately because they don’t share info; likewise the Student Loans Company even though there is a close connection between income and repayments [0]; and when I tried to tell the DVLA, they told me they lacked the capacity to know about non-UK addresses.

[0] the repayment rules means that, for low- and mid-income people, it behaves more like a graduate tax on worldwide income than a loan — it’s a percentage of pay over a threshold, and only high earners can pay it off fully — and if you do a self-assessment tax return there is a box for student loan stuff.

To vote, you need to register to vote, I voted at London Mayor elections in 2016. You vote wherever you register to vote, I don't remember the details but I guess it was me declaring that I live there and I would like to vote.

In UK there's this concept called proof of address and it's usually bill or a document that is sent to your address, there's no central registry of residents.

The point is, you don't need a register, central or otherwise, for the place of residence to be legally relevant. The way this kind of things work is that, should there be a legal challenge, you will provide a proof of residence to the court or whatever relevant body, just like you do when registering to vote.