OK, our experiences from very mixed inner-London primary schools:
Fighting, kicking or systematic bullying absolutely not okay. Playfighting also not okay. Serious physical incidents (punching, kicking etc) would result in a child being sent to deputy head/head for a bollocking. If serious enough (proper fighting, group of kids attacking another) child would be excluded. This happened only twice to my knowledge in the 10 years I had dc in that school.
Kicking a teacher would be instant exclusion for a child above nursery/reception age, as would swearing at a teacher. Again, there's only one incident that I ever heard about (and I would have, it was that kind of school).
For pushing, shoving, nastiness, the child or children would be talked to by deputy head to get to the bottom of the situation, and punishments/consequences meted out as necessary. For repeated incidents, parents would be contacted and called in. If child really can't behave, the school has the right to make parents take the child home at lunchtime, or for the child to be kept apart from others at playtime. This wasn't common, but it did happen. I think they also do behaviour contracts with the kids + parents.
We had a minor incident when ds was in Y1 which involved a large-scale playfight with obvious potential to get out of hand (think the boys were playing at 'gangs'). All children even vaguely involved were spoken to individually by the head to get an idea of what actually happened, then received individual bollockings too. We had a letter home asking us to reinforce the message at home. More recently, ds (then Y4) was on the receiving end of some overtly homophobic and very graphically sexually explicit verbal abuse from a Y6 boy. I went in to see the deputy head to complain, and I know it was dealt with, though don't know details obviously. Absolutely no recurrence nor any repercussions for ds though.
Buddy schemes tend to operate on the basis of a group of kids being trained in peer-mentoring. Some schools also have a 'buddy bench' where dc can sit if they feel lonely at playtime, and someone will come and scoop them up to join a game.
Schools also tend to have very finely-tuned reward and consequence systems, and inappropriate physical behaviour would be part of that. Lots of schools also have a playground code which the kids have usually worked on themselves and which are prominently displayed around the place. This would generally cover things like keeping your hands and feet to yourself, not excluding people or being nasty about them, not throwing things etc.
In all the schools my dc have been in there have been significant numbers of kids from really difficult family backgrounds (prison, domestic violence, alcohol, drug abuse, you name it). Physical or even verbal violence at school was the absolute exception, and all the schools have gone to great lengths to investigate incidents thoroughly and deal with them well in order to create an environment where all children feel safe. It is time-consuming, but I think most UK schools would regard it as a priority.