How long can a very newly diagnosed type 1 diabetic survive without insulin?

This is a complicated question to answer. For someone like the teenager in your question, there would be some days or months during which time the lack of insulin from a presumed autoimmune attack would make producing insulin in the pancreatic beta cells problematic. There is some period of time before symptoms show, when high sugars would be intermittent – so not many symptoms produced. As the pancreas deteriorates, the high sugar levels go up and get sustained. Eventually, this produces excess urination, weight loss, ketones from the body trying to use fat for energy since sugar and protein (that gets converted to sugar), still cannot be used correctly without available insulin. This leads to ketoacidosis and, ultimately, severe dehydration and kidney failure, lung failure, cardiac failure and/or brain complications alone or in combination. A general answer would be, for type 1 diabetes, days to weeks to months, but then would also be adjusted as to any other factors like an infection, emergency surgery, etc. The younger the person, the more quickly the usual decompensation. With type 2 diabetes, symptoms can last for months or years because there is relatively more insulin available for longer periods of time than in type 1 autoimmune diabetes but still there are variables based on other conditions, obesity, cardiovascular and blood pressure problems, other medications being taken, severity of illnesses and infections etc. So, usually months or years for type 2 and usually weeks or months for type 1.