So each of the three given lists has the property that:
it can be encoded with a mono-/digraph in context. The first list consists of chemical elements, which can be encoded with their atomic symbols. The second is a list of countries, and can be encoded with the ISO 3166-1 Alpha-2 standard. The third is a list of metropolitan areas in the United Kingdom, and can be encoded with their postcode areas.
Placing the encoded lists next to each other, we get:
1. N - NU - ???
2. F - BR - E
3. AR - ??? - M
4. P - PR - PR
5. Y - MA - M
6. U - NE - ???
7. U - LY - L
8. AU - ??? - GU
9. ??? - ??? - S
10. ??? - TO - CT
11. ??? - NO - ???
12. C - DE - DE
Each of the given codes is the first instance of a code that appears in the English month that corresponds to the position in the list. So for example, the first atomic symbol appearing in JANUARY is N, as none of J, JA, A, or AN is a valid atomic symbol. Similarly, neither of JA nor AN is a (current) ISO 3166 symbol (though both have been used historically...sneaky!), but NU is. Hence the complete lists are:
1. Nitrogen, Fluorine, Argon, Phosphorus, Yttrium, Uranium, Uranium, Gold, Sulfur, Oxygen, Nitrogen, Carbon.
2. Niue, Brazil, Morocco, Puerto Rico, Morocco, Niger, Libya, Australia, Sweden, Tonga, Norway, Germany.
3. North London, East London, Manchester, Preston, Manchester, North London, Liverpool, Guildford, Sheffield, Canterbury, North London, Derby.
A final note:
The title "This won't work with US states" is appropriate because JUNE, for example, contains no US postal code abbreviation. Except that as Stiv pointed out, it does (NE - Nebraska). But July really doesn't contain a US postal abbreviation. I think...