How fast can I lose weight when avoiding carbs and eating veggies and meat, wholegrains and complex carbs?

At loss. It depends on your wt .The heavier you are the the easier it is to lose wt. .
You don't . want to lose weight too fast. It'll just come back with interest. See a physician or nutritional therapist and get on a lifestyle modifying program that will allow you to keep the weight off for life.