Dr Marge Wissman's excellent website includes info on budgie obesity. I hope your vet has already probed for tumors in Twill. They can add appreciable weight to such a small bird

Hoping hard this is not the Twill's situation.
I know from my own dieting experience that loss isn't usually as fast as I'd like. I've tried unsuccessfully for 3 weeks to lose 5 lbs gained over the holidays. Those 5 lbs are just 3.5% of my weight. That same 3.5% of a 40 gram budgie's weight is a mere 1.4 grams. You might be looking for more loss in a shorter time than is practical or healthy for such a tiny bird.
Here's what Dr Wissman says about obesity in budgies and 'tiels:
Budgies, and to a lesser extent, cockatiels, are very prone to obesity and the problems related to being overweight. The obese budgie or cockatiel may develop lipomas, benign fatty tumors. These may be found over the crop area, the chest, or most commonly, the abdomen. In other cases, the bird may develop generalized lipomatosis, which is a layer of fat over the entire surface of the body under the skin. Xanthomas, yellow fatty tumors, may also occur. Surgery may be necessary, especially if the skin over the tumor ulcerates, but often, the tumor will recur, unless changes are made in the diet and activity level.
Obese birds usually have some degree of liver problems. When fat is deposited in the liver, normal liver cells are replaced with fat and over time, if enough normal liver tissue is destroyed, it becomes cirrhotic. Birds with hepatic lipidosis will suffer from some degree of liver dysfunction and may bleed excessively, as the liver is responsible for providing clotting factors in the blood. Hepatic lipidosis is very serious and can be fatal. Prolonged liver damage may result in the liver becoming fibrotic eventually, leading to cirrhosis of the liver. Hepatic lipidosis can have multiple causes, and may have a genetic predisposition. Birds on an all-seed diet with restricted exercise are prime candidates for hepatic lipidosis.
Thyroid dysfunction may result in obesity; however this has been rarely documented in any psittacines, and there are no reports in the literature documenting hypothyroidism in cockatiels. Toxins such as aflatoxins can result in fatty changes in the liver. Steroid administration (from topical ointments) or hormone injections (with methylprogesterone, for example) can cause a bird to gain excessive weight.
Budgies and cockatiels with liver disease may have overgrown toenails and beaks. The best indicator for liver disease in birds is an elevation in the bile acids in plasma or serum. To definitively diagnose liver disease, a liver biopsy should be performed. However, in many cases, a bird with advanced liver disease cannot withstand the stress of a surgical procedure and may have problems with hemostasis, so surgery may not be recommended or possible.
While I do not condone allowing any pet bird free-flight in the home due to the risk of injury or escape, it must be remembered that budgies and cockatiels are naturally very athletic birds, so the largest cage that is practical should be purchased. They should have toys, swings, ladders and playgyms, and owners should encourage their birds to play and exercise frequently to prevent obesity.
Obese birds, and those with lipomas, should be offered a diet lower in fat. Each case should be dealt with individually. As guidelines, I usually recommend decreasing regular seed mix and increasing millet, since millet is lower in fat than the other seeds. Sprouted seed is healthier, and should be offered daily. Harrison's pellets can be a part of a healthy diet (I use these as the basis of the diet for Quaker parakeets with hepatic lipidosis). I also recommend offering pasta and whole wheat bread, plenty of fresh veggies and fruit, and some table foods (without butter, margarine or other added fats). Increasing the activity level of obese birds should be undertaken.
To this day, some budgie books still state that feeding greens, fruit and vegetables will cause diarrhea. Of course, we all know that the consumption of foods with more water in them will cause increased urination, and not diarrhea. Feeding vegetables and some fruits is recommended and bird owners should not be put off by printed misinformation that discourages them from feeding those healthy foods.
Margaret A. Wissman, D.V.M., D.A.B.V.P.
www.exoticpetvet.net