RAI: You are free to use your Bonus Action and movement
Jeremy Crawford has given his input:
The berserker axe doesn't prevent a bonus action, and you can move freely if you fulfill the attack requirement.
I believe his Intent is different than RAW below and that's okay and it can be up to the table/DM to go with RAW or RAI.
However, you must still fulfill the attack requirement (complete all attacks) before you can use that Bonus Action freely. In your scenario, your Redemption Paladin would move to the nearest opponent, take their Attack action (includes attack and extra attack), and then you can your reaction (after the extra attack)/bonus action/remaining movement.
RAW: Bonus Action and Reaction must attack (if available)
Berserker's gonna go Berserk. The curse seems clear in that when under the curse of Berserk, all you want to do with your action choices is attack with the axe.
The Berserker Axe stipulates (emphasis mine):
While berserk, you must use your action each round to attack the creature nearest to you with the axe. If you can make extra attacks as part of the Attack action, you use those extra attacks, moving to attack the next nearest creature after you fell your current target. If you have multiple possible targets, you attack one at random.
The bolded section only states that you must your action to attack the nearest opponent. The question now is are Reactions and Bonus Actions considered actions.
Bonus actions are a type of action, as explained in the PHB (189):
Anything that deprives you of your ability to take actions also prevents you from taking a bonus action
Since the Berserker Axe requires you to use your actions to attack, then should you have a Bonus Action that is an attack, you must use it attack.
What if you don't have a Bonus Action attack?
Unfortunately, it doesn't really matter. The Bonus Action is still an action, and if you can't attack with it, you still can't use it because you must use your action to attack with the axe.
We can look to the Incapacitated condition to get some guidance here. Being Incapacitated (PHB, 290) details:
An incapacitated creature can’t take actions or reactions.
The separation of those two things suggests that they are different and that a Reaction is separate from an action.
However, the PHB (page 190) does clearly state:
Certain special abilities, spells, and situations allow you to take a special action called a reaction.
This is pretty strong evidence that a reaction is an action as well.
Note: I still think it's weird that you have your choice of Reaction here. If you're Berserk and attacking whatever is closest to you with your action, then why would the Reaction not be included in your Berserker state?
Because of that clause in the PHB about special action and the above that it just doesn't make sense thematically, I don't think you'd be able to use your reaction in any way except to attack.