I'm the family cruise editor here and we too have milk sensitivities in our family. My daughter and I bring our own soy and rice milk boxes with us when we cruise. We get the small individual servings and pack them in our carry on luggage. Many new ships have in room mini bars to cool them or you can use your ice bucket. The kind of soy and rice milk we get doesn't have to be refrigerated until it's opened.
You can also call the cruise line ahead of time and see if they what they can do for you. While I have found that waiters are pretty good at telling you what's in food when you ask them, if you need special milk you'd probably do best to bring your own.
Please let me know how it works out upon your return.
Smooth sailing,
Luisa Frey Gaynor