Let's go piece by piece here.
- Manufactured: since we are talking of food, I guess the best word would be "Préparé".
- Brewed: in your specific context, you are talking about tea, so we agreed on "infusé".
This is a hard one since it is quite vague. "Atelier" is commonly used. It does render as "shop", but hardly the selling kind. Rather a place where (manual) craft is performed (sewing shop, workshop, repair shop, ...). The atelier can be the place where the selling takes place but it is often in a separate room or building.
If you want to keep really neutral, "bâtiment" is the literal translation for "building".
Literally, this is the verb "traiter". "Utiliser" is however more common for this context.
Peanuts, tree nuts and dairy/milk products
- Peanuts: "cachuètes" or, more widely, "arachides".
- Tree nuts: there is no French term for all tree nuts. "Noix" is for "walnuts" only.
You could use "fruits à coque" ("shelled fruit") as Alexandre d'Entraigues suggested. It regroups both peanuts and tree nuts.
- Dairy: "produits laitiers".
Bringing it all together
Keeping close to standards is a good idea as it allows consumers to easily understand the sentence since they are used to see it everywhere when they are looking for allergens.
As a consequence, I would go with the following (which is close to what Alexandre d'Entraigues proposed):
Préparé/Infusé dans un atelier qui utilise des fruits à coque et des produits laitiers.
This is implied in cl-r's proposition:
Peut contenir des traces de fruits à coque ou de produits laitiers.
In the first form, we used "des", not in the second form. The additional products are countable, so we use a definite article, which contracts with "de" ("de + les = des"). In the second form, since there are only traces, they become uncountable and we use only "de", without the article.