What statutory or prerogative authority is the FSA exercising when it allows such false labelling?
As other answers have noted, they have executive discretion to "turn a blind eye" to what would normally be a violation of food safety laws and just decide that, right now, their government-mandated priorities are better served by not enforcing that particular letter of the law in these particular circumstances.
In theory, someone could probably challenge that decision and take them to court over it, claiming that they were derelict in their duties or not acting impartially and in the public interest. However, in this case they'd probably have a pretty good defense against and such claims, both due to having explicit authority to make such decisions in emergencies (as also noted in other answers), and also because the decision itself seems pretty fair (it applies to all producers using sunflower oil in their products) and justified by the circumstances (more on that below).
What public good is supposed to be being served here? It is stated that allowing items to be labelled "sunflower oil" that contain no sunflower oil "[maintains] the supply of certain food products", but this is nonsensical because replacing a product with another product does not maintain its supply, regardless of whether the replacement product is labelled accurately or falsely. Labelling the replacement as being the product it replaced appears to maintain its supply, but it doesn't actually maintain its supply. That is fact rather than opinion.
You seem to be assuming that this advice by the FAS is about allowing someone to sell bottles of "sunflower oil" that actually contain rapeseed oil. It's not.
It's not even really about products like crisps (potato chips, for those not in the UK) or pesto sauce or margarine that might contain sunflower oil as a major ingredient. For those products, the news page you linked indicates there are separate rules that apparently require, at a minimum, applying a sticker to notify consumers about the substitution:
"where sunflower oil is a key ingredient, such as crisps, retailers will imprint information on substitute oil onto existing labels."
However, think about all the other foodstuffs sold in stores that might include sunflower oil as a minor ingredient, like, say, granola, mustard sauce, instant noodles, mashed potato powder, microwave popcorn, frozen pizza, bolognese pasta sauce or just plain white bread.
(If you're wondering where the random list of products above comes from, I just looked quickly around in my kitchen for products that had sunflower oil, rapeseed oil or some other neutral vegetable oil listed as a minor ingredient. There are surprisingly many.)
I assume you would agree that all of these products would still be substantially the same product regardless of which kind of vegetable oil was used in them.
The companies making all those products, and many others besides, typically order their product packaging in bulk, often from overseas, getting a new shipment of boxes or wrappers or labels maybe every few months or years. And of course those will all have the ingredients list printed on them, as mandated by law. If they want to change the ingredients, that means they have to send a new design to the company that makes the packaging and wait however many days or weeks or months it takes for the packaging company to make and ship the new packaging with the updated ingredients list to them.
Normally that's not much of a problem, because normally food manufacturers don't tend to change ingredients in a hurry. Usually they'd plan such changes months in advance, order the new packaging well ahead of time and probably use up all of the old packaging they have in stock before actually making the switch so that they don't have to throw it away.
(Companies that do need to frequently switch ingredients, e.g. due to seasonal or unpredictable availability, usually plan for that in advance e.g. by making their labels generic enough to accommodate the change or, where that's not allowed, finding workarounds like indicating the exact type or origin of the ingredients in codes that are stamped on the packaging late in the manufacturing process. That's why you occasionally see stuff in ingredients lists like "vegetable oils (rapeseed, sunflower or soybean)" or "produced in EU and non-EU countries" or "see last letter of expiration code for country of origin: A = Spain, B = Morocco, C = Israel, D = China".)
In this particular case, however, a lot of companies that had been using sunflower oil in their products, and expected it to remain easily available, were caught short when the war in Ukraine broke out and the price of sunflower oil suddenly went way up, as companies selling the oil realized that there probably wouldn't be much sunflower oil exported from Ukraine this year.
At that point, the companies that had been using sunflower oil as a generic cooking oil in their products would normally have a limited number of options, none of them particularly good (for either the companies or consumers):
Keep using sunflower oil at whatever cost and transfer the increased cost to consumer prices. (To make things worse, the more companies do this, the higher the price of sunflower oil will rise, as they're basically competing for a limited supply.)
Switch to a different type of oil and order new packaging ASAP, hoping that it will arrive before your existing stock of sunflower oil runs out. (This might take longer than usual, since presumably other companies are also in the same situation, so the packaging makers are probably swamped with sudden orders. Also, your stock of old packaging is now useless, and you might have to throw it out.)
Order the new packaging immediately and pay extra for expedited delivery. Again, you'd be competing with lots of other companies who also really want to be the first to get their new packaging, so premiums for fast delivery are likely to be high if you can get it at all. All of that extra cost will also likely transfer into consumer prices.
Use the existing packaging but apply stickers with updated ingredients lists on top of the old ones. That's not nearly as easy and cheap as it sounds, not only because you still have to get the stickers printed (and the printing companies are probably also swamped with orders), but also because you'll have to apply them to every single box or bag or carton, likely by hand. That's a lot of expensive manual labor that will, again, likely increase consumer prices.
Just halt production until you can get new, updated packaging (or more sunflower oil at a reasonable price). That's probably the worst option for both the manufacturers and the consumers, since it results in lots of lost income for the manufacturers and product shortages for the consumers. Still, if all the other options are even more expensive, some manufacturers might be forced to do this.
The exceptional decision by the FAS to selectively enforce the food labeling requirements in this particular case basically offers these food manufacturers one more option: switch to an alternative type of oil now, but keep using the old packaging until you can replace it.
Practically speaking that's probably the best outcome for almost everyone. There's really very little difference between sunflower and rapeseed oil — both are neutral, mostly flavorless vegetable oils suitable for generic cooking purposes — and most people probably can't really tell them apart, especially not when they're used as minor ingredients among many others.
Of course, there may still be people who really don't like the taste of rapeseed oil (assuming they can taste it) or are allergic to it (which, as the news article you linked notes, is very rare) or have some ethical or religious objection to consuming it (not that I'm aware of any, but I'm sure someone out there has one). Hence the news release, so that those people who might be affected by the substitution can find about it in advance.