Most products here are now labelled as to whether they contain nuts. The trouble is that manufacturers and stores are so paranoid that just about everything that has been within 200 yards of a nut says it might contain nuts! Good supermarkets often now have sections of foods specifically designed for people with particular intolerances -- nuts, gluten, dairy, whatever.
In general nuts probably aren't used as widely in foods here as they are in the us. Milk is probably even easier to avoid. Most supermarkets sell soya-type milk substitutes, as well as goat's milk. One is even selling buffalo milk! You can also get non-dairy ice-creams, etc. There are quite a lot of vegans here, maybe that helps.
I would also recommend that you think about getting some vitamin d supplements here, or at least taking a good multi-vitamin. (it seems that vitamin d may help to regulate the immune system, and allergies are after all an excessive immune response.) anywhere in england is actually a lot further north than any place in the us except alaska, so there is a lot less sunlight, and it can be impossible to get enough vit d from your diet.