Many of the schools here unofficially decide at the start of Year 10 when they are put into sets that are intended for Foundation or Higher. It's officially still possible to move, so people don't object too much at that stage, until they realise that in practice, moving is extremely unlikely, through a combination of timetabling, class sizes, and missed material. I have tutored a few Year 10/11 who would like to learn some of the Higher content, but the set up at school means that once they are in Foundation, that's pretty much it. By Year 11, the Higher classes have done so much more work that it would be really hard for them to make up the missed content even with a tutor. And switching maths sets could mean changing other classes, and when they have a sort of 2-year program, usually with the same teacher, that can be complicated.
other schools manage it better, lots more maths sets all along, less of a fixed divide between foundation/higher classes, more reviews of sets, more students generally which can mean more maths classes timetabled at the same time, allowing moves, etc etc.
so it's quite a school specific thing. But overall I think switching from a class that is intending to do Foundation to one that is intending to do Higher, is quite an effort; if they are in a class that is unsure which tier they will do and is teaching some crossover 4-6 content, then it will be an easier move. Going the other way, from Higher to Foundation, is generally easier (unless they are so lost in a higher class that they are getting nothing from it, and missing out on learning some of the basics well, which you would hope would be noticed earlier!). I think larger schools where there are more finely graded sets can make that aspect easier.
But the sooner the better for changing to Higher, for a pupil who really wants to do it.