It is true, it seems, that Nicotine treats ADHD. There is at least one study that showed nicotine patches (18h and 24h ones) provide a mild to moderate relief for ADHD in people that do not use nicotine otherwise (e.g. nonsmokers). So, it doesn't just treat the symptoms of nicotine withdrawal.
Cigarette smoke is known to contain a clinically significant amount of MAO inhibiters that are not nicotine. MAO inhibiters are regularly used to treat depression and anxiety, and theoretically could treat ADHD since MAOI's generally prevent breakdown of dopamine. There is some nuance here.*
- For example, there are different types of MAO (monoamine oxidase - an enzyme that breaks down monoamines. Dopamine is an example of a monoamine), like MAO-A vs MAO-B, which break down different monoamines, and MAOI's (MAO Inhibitors) differ on whether they inhibit A, B or both. Furthermore, there are reversible vs irreversible MAOI.