The answer is - very well! Like I said, just about all adults are sensitive to milk in one way or another. Milk is a substance created for babies to drink. It's not meant to be drunk by adults, so most adults develop an intolerance for it as they age. It's quite normal to have a milk-free diet when you are older.
The core of a low carb diet is healthy vegetables, healthy meats and fish, and 8 glasses of water a day plus vitamins. None of these things involve dairy or lactose. Sure, some people CHOOSE to eat cheese. Some people CHOOSE to use cream or half-and-half in their meals. But there is no reason you have to do these things. You can eat healthily and well and not eat dairy products.
The main concern would be to ensure you have an adequate calcium intake. A normal multivitamin would take care of this. But other than that, you can easily substitute soy milk or other milk products for smoothies or ice cream or omelets. And other than those few things, there's really very few low carb recipes that even involve dairy products.
So in short, enjoy a low carb, healthy lifestyle! There's no reason that milk should at all be involved in your diet if you don't want it to be.
Low Carb Basics
500 Low Carb Recipes
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Lisa Shea's Library of Low Carb Books