My iron, stone, and wood tools all had damage that was consistent with the previous world. The diamond tools with about 50% damage were the only ones I could see a visual difference. However, after using diamond tools which appear to be mostly damaged, the issue may only be that the damage bar indicator is hard to see on the half damaged diamond tools.
edit - after looking at the attached image more closely, I believe that the true issue is that the damage bar is much harder to see after conversion than before conversion. On the damage bar for the iron and wood tools, the bottom portion of the bar showing damage is black, and the top portion of the damage bar contains color. On the old, the color is a larger percentage of the total height, and on the new, the color is a smaller percentage of the total height. This makes the damage bar harder to see. Also the background color and damage bar color are different between the two, and the old background colors give a greater contrast.
Perhaps the true issue is "damage bars colors on Switch Bedrock are hard to see"
Same issue for me in a Switch imported world. Sitting tamed pets cannot be commanded to follow.