If there is no family history of allergic disease, food allergies, etc then peanuts are considered ok to give from 6 months. It is only if you have a family history of the above that the advice is to wait until 3 years. My dd loves her peanut butter and jam sandwiches!!
Babycentre says the following
The government's COT report (Committee on the Toxicity of Chemicals in Food, Chemicals in Food, Consumer Products and the Environment) recommends that women who have a family history of allergic disease (or who have a partner who does) may wish to avoid peanuts and peanut products while pregnant and breastfeeding to reduce the risk of their baby developing a peanut allergy. Allergic diseases include asthma, eczema, hay fever and food allergies. It also recommends that babies from families with a history of allergic disease stay on a peanut-free diet until the age of three years.
If you have a family history of allergic disease, you may also choose to exclude all nuts from your baby's diet, not just peanuts. The reason for this is that, if a child is allergic to peanuts, they may be allergic to some other nuts. This is because the proteins found in peanuts and other nuts have a very similar structure.
If you have no family history of allergic disease, these recommendations do not apply and peanuts can be included in both your own and your baby's diet. In fact, peanut butter can be a useful weaning food. It is highly nutritious, and a good source of protein, particularly for vegetarian and vegan babies.
If you are using nuts in your baby's or toddler's diet, do make sure they are ground or processed into small pieces, so that your baby can't choke on them. (Children under five should not be given any whole nuts because of the risk of choking.)
Also says
It's true that you shouldn't give whole nuts to children who are under five years old, because of the risk of choking. But nut butters and ground nuts are fine for babies over six months old and children who do not have a family history of allergies.
Serious allergies to nuts and nut products and some seeds affect less than one per cent of the population. Your baby may be at higher risk if you, your baby's father, brothers or sisters have certain allergic conditions such as hayfever, asthma and/or eczema.
If your baby is in this higher-risk group, avoid eating peanuts and peanut products while pregnant, during breastfeeding and during the introduction of solid foods. Until your child is at least three years old, peanuts and peanut products should be avoided.