Nuts are banned (allergies)
There is a "healthy eating" policy in school, which depends on how previous the mammies in that year are as to how strictly its enforced: when DD started, a small treat for everyone in the class was allowed on DC birthdays (provided by the birthday DC) - but there was no expectation that it would happen as many didn't. But a group of DPs with much stronger views arrived when DD was in Y2, so that was the end of birthday treats across the whole school.
Nothing is outright banned, but there are definitely strong vibes to stick to a healthy sandwich (or healthy alternative), some popcorn rather than crisps, and fruit - an occasional "home baked treat" is also grudgingly allowed. However, in DD's class, while there is a generally healthy group of DCs, there are regular chocolate bars, crisps, cake etc. The teacher also has a few funsized treats in the "reward jar" (along with homework passes, mini notebooks, fun stationary, the odd pack of football cards or other pocket money type treat etc) that the DCs who earn rewards can choose. Funnily enough, most of them ignore the chocolate and will wait a few days to claim a reward if there are no homework passes available!!
I forgot - yoghurts are outright banned (mess), as are fizzy drinks (sugar levels).
But fruit squash in water bottles is fine, or the odd fruit shoot etc.
Our school is oddly liberal in ways, and not in others. The food gestapo are in the lower years though, so at least that's one battle the DPs in our class don't need to worry about (far more working DPs in our year, and with a very realistic view of the world - a few treats are ok as long as its not too frequent and certainly not daily).
Last year's teacher had a great way of getting around it for class parties - she would start with sausages and one boy used to bring in noodles (teacher would bring a flask of hot water from staff room kettle), and they would have a class breakfast with various things including hm flapjacks, hm brownies, popcorn etc and a few sweets etc - but it mostly was more heatlhy versions of things in general (and they are the senior end of school now so its almost seen as a "we're getting towards more adult type of celebrations" event by the DCs)!