Is a sprouting carrot lower in sugar and does it lose nutritional value?
The answer to this is sort-of. There are a range of different forms of sugars that are involved here.

Tap roots like those found in the carrot are a form of storage of energy for the plant.  When growing the plant generates simple sugars such as glucose and fructose via a process known as photosynthesis. These sugars are readily transportable and metabolizable. Some of the sugars are used by the plant to run its metabolism, but photosynthesis is super efficient and generally out-weighs the metabolic needs of a plant so that excess sugars are generated. These sugars are transported to the root where they are converted in to the complex sugar storage form of polysaccharides, which are very stable and can be used as structural components in the plant (cellulose generally) and many other things. In plants the energy storage is usually a polysaccharide known as starch, which does not taste sweet, and is the major component in things like potatoes. This is also why young carrots, which have not had time to convert their sugars to starch taste sweet but old carrots, which have converted all their sugars to starch already do not taste sweet.

Once the carrot starts to sprout it produces an enzyme that converts the starches back into simple sugars, which can then be used in metabolism and transported to the regions of the plant that need them to grow. Growth in this case means production of leaves and new secondary roots. The leaves will, in the normal course of things, allow photosynthesis to take place, producing more sugars than the plant needs for its metabolic needs again.

In the absence of light and subsequent photosynthesis to produce more sugars, the growth process will eventually convert the stored starches back into sugars and deplete the sugar content through the metabolic process of respiration, which takes sugars and burns them (figuratively speaking) to provide energy, releasing carbon dioxide and water. This would result in a less sweet tasting carrot over time.