First of all, what kind of a fight is this? Is this referring generally..as in we could involve Thorondor, and Landroval, Gwaihir and all the Eagles in Arda versus all the Nazgul and Fellbeasts in Middle-earth in a fair fight..like they're in a coloseum or something?..LOL, or...we're just considering the situation they were in during the Battle at the Morannon?
OK..I'm not siding with anything OK? Because they're both cool and I will kill myself if I have to choose. I'm just posting some of the great stuff that they could do for survival, "asset" as you might say.
Brains:
Eagles- though they are in Manwe's command, I'm definitely sure that they have their own mind and will. They could do whatever they think is the best and not just always "Yes, Master! Up, up, and away!"...they have their own minds to choose what they want to do. They have strategic skills and something of that sort.
Fell Beast and Nazgul- they have to go together, without the Nazgul's motivations, I doubt these beasts will go their own way fighting and stuff. They're just like Orcs and Trolls, without Sauron and the Nazgul, I don't think they will be that badass. Back to the beasts, sure they fought at the Siege of Gondor without their Masters riding on them, but I thought they had enough motivations to do that. On the plus side, I think it's also good considering their character, they don't have to think, just fight and kill....which is an essential strategy for the dark side. 😉
Speed:
Actually it's impossible to say which were faster, Fell Beasts or Eagles. Both are fast. There is some evidence to suggest that Nazgul are faster, though, namely that they were able to fly in the Battle of the Captains of the West, from the Morannon, across Gorgoroth and thence perhaps over half the way to Orodruin in less than a minute -- of course, they burnt up when Gollum finally fell in. There is the feeling that they did manage cover a lot of the distance between the Gates of Mordor to Mount Doom before the Ring was destroyed. But maybe that was also because they were driven by Sauron's desperation to get there.
I'm running out of time...have... to...finish this:
General fighting skills and abilities:
First of all, Fell Beasts plus a Nazgul riding on their backs is certainly a plus point for their side against the Eagles alone.
Fell Beasts w/ Nazgul - The Fell Beasts are probably better aerial warriors. I said probably because we'll never know, that's why we're having this debate. Fell Beasts have such big mouths, with terrorizing teeth and all that could bite you and eat you whenever they have enough chance. Eagles were dreadful to those on the ground, but such birds of prey need to drop from a height and strike with talons. The Nazgul not only wielded swords and dragonish creatures that probably could fight in the air, but fear.
Eagles - Yes, at the Morannon, the clouds of ash and the summoned elements of weather would be a deterrent to them. But I do think a squad of Eagles would give the Fell Beasts and their Masters a very hard time for their part also. 😉 After all, the Eagles do have a solid history of combat and taking care of business. Remember, these were the creatures that fought and destroyed Ancalagon and his brood in the War of Wrath. Even if the Eagles of the Third Age were one-third the size of their kin from the First Age, they would still have a wingspan of roughly 50 feet along with talons, beaks, brains, and the ability to fight as a team in battle.
You decide. 😉