A good school will try to leave the decision as late as reasonably possible because students do progress at very different rates. At our place we have 5 or 6 sets depending upon the year group.
Sets 1&2 all end up doing higher and are taught the full higher tier course.
Set 3&4 are those that fall somewhere in between - too good for foundation but not quite able enough to access the full higher tier course (for example they may just about get trigonometry but trying to do it in 3D would be a step too far). They cover a scheme that is basically a reduced version of higher tier with those who fly doing the higher tier in the end and those who don't doing foundation.
Set 5 all end up doing doing foundation tier.
Set 6 (when it exists) is very small and is for those students for whom full foundation is a step too far currently. They may do qualifications below the level of GCSEs or do GCSE but really aiming at F/G grades.
The decision of higher/foundation is usually made in Feb after the mocks but some final changes can be made right up until the day of the exam (I can't recall a decision ever being made that late).
As for dual entering higher & foundation with one exam board - you can't do that as the papers include some 'overlap' questions that are on both foundation and higher tier papers.
Although the maths exams for different examboards are usually on at the same time you could, in principle, do Foundation with EdExcel (say) and then immediately do Higher with AQA or OCR because you've not left the exam venue and therefore can not have heard the questions from people in other schools who sat it at the right time. The school can even give you a short break if you are kept in a secure location without access to the outside world and supervised carefully.
It's not an approach I support but it is technically possible and some schools have played that game to boost results previously.