I would probably give my almost 3 year old a whole almond - I haven't yet but I never really thought of giving her one. Not sure about peanuts or smaller nuts. I do give my 14 month old ground nuts or nut butter and waited till she was a year for that and she hasn't had peanuts yet.
For allergy reasons, you usually start nuts (not peanuts) between 1 and 2 years - peanuts between 2 and 3. For young children, nuts should be ground or minced. Depending on your child, they can start whole nuts when they are chewing well - 2 1/2 to 3.