In my dojo we begin doing weapons-practice from the first day we start. This includes suburi-practice, jo-sabaki, to-sabaki, ken-tai-ken and ken-tai-jo. Off course we exsercise extrem causion when doing paired weapons-techniques with beginners, but then again, I guess we should allways be extremely causius when practicing
The special Nishio Sensei style of Iaido (Aiki-to-ho) is taught as well. Though it is not mandatory - it is highly encouraged.
I have done no other style of Aikido, so I am not going to try to judge whether or not 'correct' aikido (if there is such a concept) is possible to acchieve without weapons practice, but I am becomming more certain all the time that becomming really good at the style of Aikido which Nisho Sensei teaches, requires getting to know the jo and the bokken as weapons and exploring how the movements of (his) aikido has developed from the way of the sword.