I don't think Italeri has anything to do with Nexus products, but I could be wrong.

As 0nn0 said, casting small parts would probably be the best solution. Toothpicks would be kind of hard to attach permanently to plastic.

I had been thinking of getting some Nexus Egyptian chariots, as my childhood Atlantic originals have long since gone to the Land of the Dead, but after hearing about your problems I will pass!

The Atlantic/Nexus chariot is not very good. The Caesar* one is much better with a wide axle and correct rear positioning of the axle and room for the two crew provided. I prefer the Atlantic/Nexus horses so I would combine the sets an end up with something satisfactory.
This company did some better wheels for both chariots but intended for the Atlantic /Nexus one. Sadly they say they are out of stock but at least it gives a good idea of how the chariots should look.
http://www.dreamsstudio.com/baueda/specials.html


David
*set H024 has two chariots