I've lost weight twice on the Mumsnet Low Carb Bootcamp while breastfeeding. The first time was losing baby weight plus years of weight gain in 2013 after my third child was born. I lost 30lb in five months to reach the middle of the normal BMI range. The second time was just losing baby weight (I have maintained a normal BMI since 2013) after my fourth child was born.
You don't actually need any carbs while breastfeeding but by the precautionary principle I suggest following the Bootcamp Light rules in which all your carbs will come from low carb fruits and vegetables, dairy, nuts and, if you are vegetarian, small amounts of pulses, tofu or Quorn.
It's important to maintain good nutrition while breastfeeding since the baby gets first dibs on vitamins and minerals. I recommend taking a good breastfeeding supplement such as Pregnacare. Many people, whether breastfeeding or not, find they need extra electrolytes in the first few weeks of going low carb. You can buy electrolyte powders or solutions, the High5 Zero electrolyte tablets used by cyclists, or make your own electrolyte drink from water with a pinch of salt or LoSalt (which contains potassium as well as sodium) and a squeeze of lemon juice (bottled lemon juice is fine).
If your baby is still breastfeeding frequently through the night, then the adaption to low carb in the first few day can leave you feeling pretty tired. I suggest having a snack on your bedside table if you feel hungry during the night or first thing in the morning.
I'm happy to answer any questions people have.