SIDS advice varies slightly all over the world, this is just because of how data analysis works really - the raw data doesn't give you a clear answer of this is risky, that is not - it's just numbers. So it's up to data analysts to look at these numbers and translate it into advice that an actual human can follow. That process itself is a bit subjective - what one person will consider is a reasonable risk, another person will consider isn't. And in addition to that, what one person considers a significant factor will differ from what other person thinks - that's when they're looking at the same data. Sometimes it's obvious like if something has a bigger impact vs a smaller one, but if the numbers are similar then it can be less clear. And when you're writing advice, you also have to consider what's reasonable for people to do vs what would be a massive pain in the arse and therefore what people are actually likely to follow.
So the data doesn't tell us at what age it's safe to move to another room, because the age of babies wasn't controlled for in the studies, just whether they were in the same room or not. I would guess this is where they've come from with both of those recommendations:
12 months: The risk of SIDS exists up until 12 months, therefore the recommendation ought to exist until 12 months. The data that we have didn't differentiate between ages of babies, so the risk reduction most likely applies at all ages.
6 months: The majority of SIDS cases occur before this age, so having the guideline to this age will make the most difference to the most babies. UK houses are also often a lot smaller than US houses, so parents are liable to want to move a baby out of their room ASAP. If people find a guideline impractical, they may well ignore it completely - by saying 6 months, most people get to about 4 months at least, which covers the most risky period.
From what I've read: The risk of SIDS drops sharply at 4 months, to the point it probably doesn't matter quite as much about being strict with the regulations, but you should be more careful about anything that contradicts SIDS recommendations earlier than this. The risk of SIDS peaks at 2 months and is highest under 4 months. After 4 months it drops to very low, so when you're talking about "double the risk" it's like 0.001% to 0.002%, something like that.
Emily Oster Crib Sheet, or there's a new one called Bottom Line for Baby (can't remember author) are really good for picking apart this stuff!
There's another good sleep/SIDS recommendation example, that I have noticed recently as a difference between countries one. The UK has somewhere in one of the official advice bodies a small line which explains that all sleep must be in the same room as you up to 6 months for all sleeps, which seems to explicitly mean that you shouldn't put them into the cot upstairs in the evening, or for a nap in a separate room. This was not always there - it DEFINITELY wasn't there in 2008 when DS1 was born, which I appreciate is ancient history in terms of safety recommendations. But it's only appeared some time in the last few years, and MN (as a community) has taken it really seriously, berating anyone who advocates for an upstairs bedtime for babies under 6 months. Now in other countries (or at least, the US and Germany which are the only two I've looked at, I'll be fair!) it doesn't explicitly say this, just that the baby should share your bedroom, which is how the UK advice used to be worded as well.
Now, again, I would guess that the data analysts have gone weeeeell... SIDS doesn't just occur between the hours of 11pm-7am. SIDS can occur at any time during any sleep. We probably should be reminding parents that SIDS guidance needs to be taken seriously for all sleeps, not just for that big chunk of sleep you (hopefully) get at the same time. And actually I think it is Lullaby Trust which advocates for this, and their main purpose is as a support charity (prevention/awareness is secondary) so it's quite possible that the change in wording was prompted by a parent who did suffer a loss unfortunately during that "bedtime" hour of 7pm-11pm or during a daytime nap where their baby was sleeping in another room, and perhaps they had taken the room sharing advice literally rather than thinking about how SIDS advice really applies to every sleep. So I can see why it would be advised to do this as well, but again, nowhere else in the world (and I don't think even the NHS) are so specific about every sleep, the wording is more just "in your room".