You can pick up an axolotl in a bucket, even when it has a lead attached. The lead disappears but will reappear when you use the bucket to place the axolotl.

If the lead is also attached to a fence when you pick up the axolotl, the lead knot remains visible but the lead still disappears. If you later place the axolotl bucket near the fence, the lead reappears and is still attached to the axolotl and the lead knot. If you instead place the axolotl bucket beyond the maximum lead length from the fence, the lead is apparently deleted from the world, because it doesn't drop at either end. Also, you can attach a new lead to both the axolotl and the fence.

Steps to reproduce:

1. Find or build a shallow pond at about 10 blocks wide or more.
2. Place a fence in the middle.
3. While standing in the water near the fence, pawn an axolotl.
4. Use a lead on the axolotl.
5. Use a water bucket on the axolotl.

Expected results: You cannot pick up the axolotl while a lead is attached to it.
Actual results: You pick it up and the lead becomes invisible, however the lead is still in your inventory. You know this because if you use the axolotl bucket on a block or in water, the lead reappears and is still attached to the axolotl and your hand.

Original description:
When an axolotl is attached to a fence with a lead and is picked up in a bucket without removing the lead first, the game thinks it still has the lead, I suppose. The lead will still appear on the fence, but is visibly attached to nothing. If you place the axolotl back down near the fence, it will reattach, but if you place it further away, the lead will disappear from the fence and will drop from the axolotl. But, if you try to interact with the lead on the fence before placing the axolotl down, the lead won't drop, and I'm pretty sure there's no way to get it back. Also, as far as I can tell, you'll never be able to put a lead back on the axolotl afterwards. I haven't tried leaving the world and coming back yet, but I doubt that will fix the issue. I could be wrong though.