Used to live there. But was in the South East, so a different take on the weather and memories. Let me see what I recall:
January: Dismal, rain, freezing, after-Christmas sales bonanza and very wet streets. Second half of the "Winter" term for schools (around 6th Jan last I remember).
February: Wet, just wet. Also, Chinese New Year celebrations in a wet cobblestoned China Town, London. Great fog banks on the motorways driving to and from London late at night (think 11pm onwards).
March: Wet. Spring. Apparently, but still brisk with some sun peeking through. However, lots of grey clouds and grey days, still. And for school, IIRC Spring Term starts around here. Somewhere.
April: Wet. Just wet. Oh, and April Fool's as well as Easter. If you're attending a Catholic school, lots of things to do with Easter. And the Spring Break (about 2 weeks of holiday, maybe.)
May: May Day. And the days are getting sunnier. School is a pain in the neck -- long and boring.
Exam prep, too. Oh, days are starting to stay sunny for longer.
June: Exams begin for many schools; some probably started in May. Days should be pretty warm at this point, little rain. Lots of indoor time, however. Usually spent at a forlorn wooden desk that rocked thanks to uneven legs, set in a large hall filled with its siblings. Pupils at a desk each. Chairs were plastic, uncomfortable and had metal legs. Shift wrong and the entire thing scraped along the wooden floor (think rusty nails on a blackboard).
July: Depending on school and district, exams were still going. By this time, the hall would have been stifling -- stuffy, hot, airless, no air-con (well, at the time I attended). And everyone fidgeting. Pencils and pens dropped, calculators clicked, one student in the back of the hall on a laptop (he had a medical condition that prevented him hand writing anything for any length of time over 20 minutes, I believe). Plenty of huge sighs (usually meaning: OMG, when is this going to end?!)
Sports Day prep was in here somewhere, too. Maybe? I don't recall the exact month. But it always took place out on the sports field -- three-legged races, sack races, spoon/egg races, etc. Parents were invited to come and watch and participate. Also a good time to make the teachers participate (and laugh at them only in the way kids can do nastily without being nasty).
Outside, sunny hot days. Sweltering compared to the long cold months. And lots of greenery (again, depends on your location). Then plenty of traffic. Enough uphill/downhill walking, that you
really didn't need the PE (Physical Education) lessons.
If you took the right route, you'd end up walking through a gorgeous park: tree canopy walkways, grassy banks with lots of tree shading, flowers, birds, plenty of bees and curious wasps. General mix of pedestrians, but quiet and peaceful.
Take a different route and you'd go through the High Street and then the actual shopping district. Push chairs. Screaming babies. Yelling adults (they were conversing, in fact). And huge gangs of students walking mostly abreast, so they managed to occupy the width of a street without having that many people in the line.
There was a huge Wimpy's on the corner (later it turned into a BK; don't know what it is now). And plenty of people hitting McDonald's at the other end of this district (which also housed another school). So, you'd see a vast amount of pedestrians holding a McD's milkshake or coke.
August: Summer holidays started and advancing rapidly. Hot sweltering days. Not much different to July, except for the absence of exams. Summer homework if you were unlucky enough. But it generally meant you'd completed the school year and you'd be advancing to the next one up come September.
Holidays abroad also defined August for me.
September: New school year. New books. New teachers (almost). New stuff to remember. Days cooling off. Long autumn sunlight; still stuffy and hot.
October: Getting cold. Mock-exams prep. Hallowe'en coming up (if you were into it). Occasional gale-force winds.
November: Cross-country idiocy. Nope, I didn't like it. It was a long job from the school, round various areas and full of migraines thanks to the cold. Mock exams prep. Sometimes these mocks took place in Nov, sometimes in Jan just after the Christmas holidays.
Rain starting up again. Gale-force winds, or the good old wind chill took up residence again. Dismal grey days once more. Nothing of interest but more homework, short days and even shorter tempers.
December: Christmas prep (for Catholic schools this was intense, almost). More studies, more rain. More homework.
Christmas holidays (3 weeks, most of which was spent having fun; mock-exam revision was crammed into the last 2 days prior to returning to school
).
Hope that helps some. It's mainly YA-focused since the original post mentioned school, summer holidays and uniforms.