After brushing your teeth, eventually you will have to eat again. And of course you can/may eat, for example nuts, after brushing your teeth. It depends on the timing. And of course: the way this question is phrased it is more along the lines of "Is it a good idea to brush teeth, then eat nuts and then go to sleep?"
Then the answer is a definite no.
My friend the dentist used to say: "A clean tooth is a healthy tooth"; so that commercial is not as bad as advertisements usually go.
Sugar is just a subtype of carbohydrates and carbohydrate digestion actually starts in your mouth. The starch amylose from potatoes for example is broken down by the enzyme called amylase found in your saliva into maltose. This a disaccharide like common table sugar and food for acid producing and cariogenic bacteria.
The amount of carbohydrates contained in the things you eat varies, but is very seldom zero (especially if the food is somehow prepared/processed).
Starches, especially processed ones, tend to cling to your teeth and prolong the time the bacteria have an ample supply of food to produce cariogenic acid and multiply themselves. That means bread after brushing is a foolishly stupid idea.
Since you will not be able able to avoid any and all carbohydrate intake for eternity it is already self-evident that never brushing again is not the smartest choice available. But aside from tooth decay caused by carbohydrate metabolising bacteria (that you now have 'avoided in theory') there is an additional aspect that will come into play even if total 'no-carb diet' was to be followed through:
Dental calculus or tartar buildup (From: "Supragingival Calculus: Formation and Control"):
Dental calculus is composed of inorganic components and organic matrix. Brushite, dicalcium phosphate dihydrate, octacalcium phosphate, hydroxyapatite, and whitlockite form the mineral part of dental calculus. Salivary proteins selectively adsorb on the tooth surface to form an acquired pellicle. It is followed by the adherence of various oral micro-organisms. Fimbriae, flagella, and some other surface proteins are essential for microbial adherence. Microbial co-aggregation and co-adhesion enable some micro-organisms, which are incapable of adhering, to adhere to the pellicle-coated tooth surface. Once organisms attach to the tooth surface, new genes could be expressed so that mature dental plaque can form and biofilm bacteria assume increased resistance to antimicrobial agents. Supersaturation of saliva and plaque fluid with respect to calcium phosphates is the driving force for plaque mineralization. Both salivary flow rate and plaque pH appear to influence the saturation degree of calcium phosphates. Acidic phospholipids and specific proteolipids present in cell membranes play a key role in microbial mineralization.
This is problematic because they are Dental biofilms: difficult therapeutic targets (10.1034/j.1600-0757.2002.280102.x):
Periodontal diseases are infections caused by micro-organisms that colonize the tooth surface at or below the gingival margin. While these infections have many properties in common with other infectious diseases, they exhibit unique properties conferred by their site of colonization and the nature of the en- vironment in which they reside.
Never brushing again will certainly shorten your life you live in comfort, regardless of how extreme you try to adapt your dietary ideas. Caries, periodontitis and other sorts of inflammation and infections would be very likely to increase.