Heya! First, the easiest question. Yes, we wear uniforms! They vary depending on the kind of school, so bear with me. Also, this is a loooong post, sorry.
Primary school = age 5-12 - Reception & Years 1-6
Primary schools can be plain old Primary schools, religious (CoE/Church of England, Catholic) or private. Most religious primary schools are also government-run, esp the CoE ones, as that is the official religion of the UK. Uniforms can look like this - summer, winter, private school
(Pre-)Prep Schools = age 7-13 ish.
Prep schools are a dying breed, and are really expensive private schools that prep (ha!) children of very rich people for Public Schools (see below). All prep schools are different and if you ever meet someone who actually went/goes to one, you’d need to ask them what years their prep school took and so on. Uniforms look like this.
Secondary school = age 12-16 (or 18) Years 7-11 (or 13)
Secondary schools can be comprehensive schools, grammar schools, state schools or private schools. They can also be religious or secular. Comprehensives are what we call plain old government-run, catchment-area schools. Grammar schools are also government-run, but they require an entrance exam to enter and are generally not catchment area (and therefore tend to do slightly better in league tables as they get to pick who attends).
Public schools and Private schools both have fees. Private schools are schools that charge students to attend, or have scholarships. Public schools also charge students, but they are the really old, really posh ones, back when ‘public’ meant that the school was open to anyone who could pay the fee (back in a time when the truly rich had private tutors). Eton is a Public school.
We also have Academies and Free Schools which get more complicated but we’ll leave that for now… Uniforms can look like this (posh Public Schools will almost always have the blazers, but otherwise the styles are interchangeable for the other types of school) -
College/Sixth Form = age 16-18 (Years 12-13)
It’s basically the last two years of education. (Bear in mind that Eggsy is old enough that mandatory education was only up to age 16 then and he might well have left school with only GCSEs. Mandatory education is up to 18 now.) Many secondary schools have sixth forms attached to their schools, and is a name from the era when secondary school years were called First Form, Second Form, etc.
College is what you call it if you leave Secondary School or your Secondary School has no sixth form attached to it and you attend another institution for Years 12 and 13.
University is where you go to get a degree. Most usual degree courses are three years in the UK, and you never ever ever call 90% of them college, unless of course your university is separated into colleges (such as Oxbridge colleges, or University of London colleges).
ETA: This is only school in England. Scotland has a completely different Secondary education system.