I don't know of any RAW that supports this, but I would rule that the Paladin flees until the effect ends simply because it sounds right to me.
This has come up a couple of times in my current game, as one of the PCs is a Paladin, and one of the main enemies is an Antipaladin. The aura specifically states: Creatures that are normally immune to fear lose that immunity while within 10 feet of an antipaladin with this ability., so therefore once outside of 10', the Paladin is once again immune to fear, neutralizing any existing fear effects. To rule any other way would be the same as ruling that when you exit the radius of a Solid Fog, your movement speed is still reduced by half, even though you are outside the fog. A more extreme example of why this isn't the case is because if it were, then that would mean that if a single space you move through is considered "difficult terrain," then for your entire turn, you are considered to be on difficult terrain.
So in short, yes. The paladin is no longer feared once outside 10' of the antipladin.
I dont think becoming immune makes you ignore things already effecting you. Isnt there a rule that immunity is like unbeatable SR? So if you are already afraid, unbeatable SR wouldnt make you unafraid.
I'm not finding anything in the Core Rules that says that, but if you wnto to rule that Immunity to something works like unbeatable SR, check this out:
Against an ongoing spell that has already been cast, a failed check against spell resistance allows the resistant creature to ignore any effect the spell might have. The magic continues to affect others normally.
According to that, the SR (Immunity to Fear) was not in effect when you were in the radius of Aura of Cowardice, but upon leaving the radius, you do have SR, so then wouldn't this rule apply?
No, because you don't get SR to shake off something already on you, you get it just to ignore it flat out the first time. It is the same as if clerics cast Spell Resistance on you after you are affected by a spell, it protects you from later spells but not from anything already on you.
Okay, I've been looking through for the rules you're referencing. The specifi spell "Spell Immunity," and it's improved versions (greater, communal, etc.) are the only reference "Immunity acts like Spell Resistance." This is detailed in the spell's description here:
The warded creature is immune to the effects of one specified spell for every four levels you have. The spells must be of 4th level or lower. The warded creature effectively has unbeatable spell resistance regarding the specified spell or spells. Naturally, that immunity doesn't protect a creature from spells for which spell resistance doesn't apply. Spell immunity protects against spells, spell-like effects of magic items, and innate spell-like abilities of creatures. It does not protect against supernatural or extraordinary abilities, such as breath weapons or gaze attacks.
Note, that this doesn't talk about immunity as a whole, only in regards to this specific spell. Even so, the line seems superflous, maybe just to add an extra level of understanding, I don't know. Whatever it is, the line itself isn't entirely needed, nor does it state that you can't stop an ongoing spell effect with it, specifically. As for Spell Resistance not being able to deter ongoing effects, this line is from the Spell Resistance entry in the Corebook Glossery:
Effect Spells: Most effect spells summon or create something and are not subject to spell resistance. Sometimes, however, spell resistance applies to effect spells, usually to those that act upon a creature more or less directly, such as web.
Spell resistance can protect a creature from a spell that's already been cast. Check spell resistance when the creature is first affected by the spell.
This is the closest that it comes to explaining about whether or not you Spell Resistance can deter a spell that is already affecting you. The bolded line states that Spell Resistance can protect from a spell that's already been cast, but then is immediately followed by a sentence saying to check SR when the creature is first affected by the spell. All-in-all it's a pretty murky subject.
All that said, nothing in either entry explains that immunity as Spell Resistance, except in the case specifically of "Spell Immunity" spells, meaning it doesn't actually have any bearing on paladin/antipaladin fear immunity conflicts.