I'd think it's much less of a concern than in space. For one, space robots are far enough away that there's a big time delay to Earth communications, these robots will have control stations <1km away. So the only electronics on board the actual robot should be fairly rugged motor drivers and wireless comms. Add in an extra redundant copy or two of all the IC's (and possibly enough motors to give redundant degrees of freedom), and a policy of pulling out and rebuilding the robot when a certain number of non-function-impairing radiation damage events occur, and it's a non-issue.
Radiation causes huge headaches in space because there's no way to pull it out or send in a replacement, and comms have seconds - minutes of lag. These don't apply to Earth based robots, even in super hazardous environments like this.
Also they won't need to lift the robot's mass into space (which costs on the order of $5000/kg), so for most of the 'bots, they will be able to easily add additional shielding around critical electronics. Robots which are intended to fit into very small spaces might not have this option, but also hopefully will be cheaper than their larger brothers.
I wonder how many robots will become part of the contaminated decommissioning waste. Perhaps they will develop robots to disassemble and store the damaged and contaminated robots, filling a function like decomposers in nature.
Radiation causes huge headaches in space because there's no way to pull it out or send in a replacement, and comms have seconds - minutes of lag. These don't apply to Earth based robots, even in super hazardous environments like this.