Interestingly, A-levels (whether completed at a college or 6th form) are equivalent to the first year of US university education.
So, a British person moving to the US between A levels & university will enter into the 2nd year of American university rather than the first.
On the point of primary & secondary education, we actually have two difference school systems in the UK.
One is:
Primary school (reception, then years 1-6)
Secondary School (years 7-11)
College (a levels) (years 12-1r)
The second is:
Primary school
Middle School (year ? to year 8)
High school + 6th form (years 9-13)
I don't know at what age middle school starts because I never went to one, but it takes children through to year 8.
High school starts in year 9 and in my experience will generally include 6th form. Regardless of whether students are in a college of 6th form, you may hear them refer to themselves as being in years 12 or 13.
(6th form is called such because it's a hang over from my parents generation and before when instead of years 1, 2, 3 etc academic years were called 1st form, 2nd form, 3rd form etc.)
Until recently, compulsory education finished after GCSEs, which is why there's a division between secondary education & college/6th form. Only students choosing to go onto university would generally take A levels in years 12+13.