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1. Send your photographs to Italeri and ask for replacement.
2. Copper wire might be a solution for the drawbar. You can easily bend it to the right shape. Then remove the rest of the short drawbar, drill a hole in the front of the chariot and glue the copper wire in its place.
(Personally, I would make a small mold and cast some resin or pewter spare parts…)
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.
*set H024 has two chariots