The key point is this: just as Argentine tax law defines the rules for when an individual becomes a tax resident, loses tax residency (and then regains it having lost it), so too does UK law. And both these sets of rules operate independently of each other: you don't automatically lose tax residency in one jurisdiction by virtue of having gained it in the other. So, the first step is to discover at what point you gain Argentine tax residency, which will be the date you obtain PR. Now you also need to discover what UK tax law says about losing UK tax residency. Once you've done that discovery, you may well find that, for a time at least, you are still going to be a UK tax resident even as you are simultaneously an Argentine tax resident. And while that situation continues (or starts again in the future, having earlier ceased for a time), it is the tax treaty's provisions (not the respective taxation laws) that determine where you pay your tax and the rate at which you pay tax. That's why playing by the book means playing by the book not just in Argentina, but also in the UK. You need to know your obligations in both countries in the new taxation situation you have put yourself in by obtaining PR in Argentina. So, you'd best start with getting advice from the UK, given that you currently know all you need to know, for now at least, about your looming situation in Argentina.
Here is the UK-Argentina tax agreement:
https://assets.publishing.service.gov.uk/media/5a7f803a40f0b6230268fdf5/argentina-dtc_-_in_force.pdf
Do not try to read one of these on your own. They contain a lot of words and concepts whose meaning is different to ordinary usage and can only be understood by reference to the overarching OECD taxation policy documents. I tried for years to understand the one that applies in my case, only to find that when I finally paid for advice, I had misinterpreted important parts.
(Be aware also that by talking to an international tax adviser, you may discover strategies now or in the future to take advantage of the UK-Argentina tax treaty to lower your overall tax liability.)