Villagers do not harvest carrots or potatoes. I have set up multiple forms of farms with villagers to harvest carrots and potatoes which they do not harvest. The only way I could get the villagers to harvest carrots was by filling the villagers inventor with beets (I assume bread / wheat would also have worked)- EDIT - That is not consistent full inventory does not always cause the villager to harvest. However, if the villagers inventory is full of beets they obviously cannot pick up the carrots to then replant.

If spawned (thus empty inventory) in a field filled with carrots or potatoes they will stand at their work station until bedtime at which point they go to bed, repeating that each day. They never harvest.

On occasion I have seen them share food to another villager but have not been able to find a pattern to when they will share. 

Tested with:

mob griefing: true

mob spawning: true

entities drop loot: true

tile drops: true

mob loot: true

The attached image is a 3 stack 9x9 carrot farm which has never been harvested even after hours of waiting for the villagers to farm it. Each layer has a bait villager where the farm should share their carrots, however, even when given beds and extra beds the villager neither harvests nor shares.

This bug seems similar, but not identical to MCPE-44268