My master's degree last year was almost completely individual research, so was probably not similar to what you'll be doing, but as a PhD student now in the US, I can hopefully give some info on the way things work in the UK vs the US.
In general, I have found that US grad school is much more demanding and focused on individual work than the UK, in terms of: (in science, at least) you are accepted on a US masters program and then you form an individual masters program of study and research project and then it takes however long to finish (you don't always have a specific end date), but in the UK, everyone on the same degree course will usually take the same classes (with the exception of some electives), everyone takes the same amount of time to complete the course and usually everyone graduates together.
I've found that the US demands a lot more work and teaching responsibilities from grad students compared to the UK (in the UK, TAing is usually only done by PhD students and it's for extra pay on top of your research grant, so is not a necessity to gain funding). The first term of my masters in the UK, I had about 4 hours of classes per week, a couple of research papers to read and no real responsibilities (I was only on campus 2 or 3 days a week and the rest of the time just hanging out at home doing not much at all) - although the classes weren't graded as the entire degree was awarded based on the thesis, so not much pressure there.
But the first semester of grad school here, I've had to teach a lab class each week (on my own), help out with a 101 class, grade over a hundred quizzes/labs per week, take my own classes every day of the week, do very difficult homework for those classes, and work on my own research project at the same time!
In terms of UK class format, there is generally much less emphasis on participation than in the US - class participation/attendance doesn't count towards the grade and so not so many people ask questions (at least at undergrad level anyway). In lectures, it's more a case of sitting quietly in the lecture hall while the prof talks at the front, but if you have seminars/discussion groups you will be expected to contribute.
Professor availability: Well, I never usually contacted my professors outside of class time, so I'm not too sure - 'office hours' don't really exist as such (as in profs don't usually have scheduled office hours), so if you want to see a prof outside of class, you would probably email or call to arrange a time to meet or just stop by their office to see if they're free.
Writing: I have no idea - I was a physics undergrad and an earth science postgrad, so apart from my undergrad dissertation and my masters thesis, I didn't really do much writing (almost my entire undergrad degree was Math - I only wrote 5 papers in 5 years)
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Socialising: I had a great social life on my masters course - everyone on the course was working in the same 'office' so we spent a lot of time together (including a month-long field trip in Central America!) - I met some of my best friends while doing my masters
. The pub will probably feature a lot in the social life of masters students
(my department had a weekly 'happy hour' on Friday at 5pm where all the grad students and profs could meet in the coffee lounge in the department for a few beers/glasses of wine and a bit of socialising - often followed by an evening in the pub afterward). There's also nightclubs, bars, the cinema, the theatre, restaurants, shopping etc. - all the usual stuff
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