Slinky you will lose weight on a low calorie diet. However there are three issues: (1) you will be hungry which makes it impossible harder to stick to the diet: (2) your metabolic output will decrease (metabolic rate, thermic effect of food, non-exercise activity levels, thermic effect of exercise all decrease); and (3) repair and maintenance will slow which, among other things, reduces muscle mass. So even when you are losing weight you will lose it at a lower rate than expected for the amount of calories you're taking in.
Then when you go back to a normal calorie intake: (1) as you've been hungry for a long time you may overeat as a compensatory mechanism; (2) it takes time for your metabolic processes to go back to normal during which weight gain may occur; and (3) regain of lost muscle mass and repair of tissues often results in a necessary weight gain which can be frustrating if you've reached your target weight or still have weight to lose. When we look at the Biggest Loser contestants they had to keep their calories much lower and their exercise levels much higher to maintain their weight loss when compared to someone of the same size who had never been overweight.
Contrast all that to losing weight on low carb where we have: (1) no need to be hungry; (2) no metabolic slowdown, in fact there seems to be a slight increase termed the metabolic advantage; (3) no reduction in muscle mass or repair/maintenance since the body's calorie needs are being effectively met due to the low insulin levels. Trying to combine low calorie with low carb can be done of course, I just don't see why you would want to. Weight loss will be slightly faster but probably not enough to make up for the disadvantages unless you need to lose weight very rapidly, for example if needed for an operation or your weight is immediately threatening your health.
The Blood Sugar Diet and the Newcastle diet (which uses shakes) are both ultra low calorie approaches, aka starvation diets, designed to reverse diabetes by mimicking the effect of gastric bypass surgery. It was observed that patients with diabetes often experienced a reversal of diabetes after surgery and Professor Roy Taylor of Newcastle University started experimenting to replicate the effect without surgery. Michael Mosley's Blood Sugar Diet is based on Taylor's research.
However you can still reverse diabetes on a low carb diet, especially if you incorporate fasting since both of these improve insulin sensitivity. Jason Fung is a Canadian nephrologist whose Intensive Dietary Management program treats type 2 diabetes with a combination of low carb and intermittent fasting - he has a lot of content on the Diet Doctor website.
All of that said, the best method of weight loss is the one that you can stick to. A lot of people like the simplicity of Slimming World or Weight Watchers and they can be easier to fit into eating out. Others like 5:2 because they only have to restrict food on two days a week which can be psychologically easier to manage. Bootcamp is a conventional low carb diet but you also get variations with carb cycling such as Carb Nite which is designed for bodybuilders and allows one carb-heavy binge per week. Intermittent fasting can be a useful addition to low carb but is by no means necessary if you don't think it will work for you. There's more to it than just weight loss of course, in my opinion the health advantages of eating low carb are significant. Not just escaping from that hangry feeling but my IBS has cleared up and I don't get period pains any more. It would be worth eating this way just for those benefits, even if it didn't control my weight.
Any questions peeps?