According to the interviews on the Wizard of Oz DVD, he was not allergic to the silver paint. The make-up department had been using silver powder on him. He inhaled the powder and spent weeks(?) in the hospital recovering. The powers-that-be realized the powder was to blame and switched to the paint.
Once they made that switch, they immediately found someone to take over the role and they never told Buddy's replacement what had happened to Buddy.