Alright, here's what I got:
Top tier:
Babylon, England, Ethiopia, Inca, Korea, Maya, Poland, Siam, Shoshone
Upper Tier:
Austria, Arabia, Aztecs, China, Egypt, Mongolia, Morocco, Persia, Sweden, Zulus
Mid tier:
Brazil, Carthage, Celts, Germany, Huns, Iroquois, Portugal, Rome, Spain, Venice
Lower-mid tier:
America, Assyria, Byzantium, Greece, Indonesia, Netherlands, Ottomans, Russia, Songhai
Bottom tier:
Denmark, France, India, Polynesia, Japan
How about Babylon? Top tier UA, but a mid tier UU that depends on very early war and putting off construction and has nothing kept on upgrade (but if you plan for early war...), and a bottom tier (by the definition of highly situational) UB that is unlikely to make the difference between losing a city or not.
Babylon's UA is more than top tier. It's an absolutely ridiculous, and snowballing, science advantage once it gets going. The UU and UB are there to give Babylon a little more reason to expand in the beginning than they otherwise might and to help them get away with single-mindedly racing up the tech tree towards Education a little more than the average civ can. The UU is halfway between the power of an archer and a composite bowman. It means that the units you do build to protect your early expansions go further, and it means you can kind of use your archers as if they were poor man's composite bowman. That's very good when you consider how many techs it takes to build composite bows and none of them are on the way to Philosophy. The UB is semi-situational but it definitely lets you take more risks with your early expansions. It adds as much HP as a regular Walls, Castle and Arsenal put together, and it's cheaper to build than regular walls (which actually are situationally useful). I think that's not bad at all, but if you are sure that you definitely will not lose the city regardless then it's obviously worthless.
BUT, the UA is so strong that it completely overshadows the other uniques, really. Even if you never build archers or walls, Ingenuity will put you far ahead of the pack. I think this is pretty uncontroversial.
We'll have to disagree on Poland, but I'm glad you brought up start biases because I consider Poland's to be excellent in general but especially for their UB and UU. You should not have trouble using the pushback ability on the WH. Add a great general if you really need one. There is no reason to get this unit too late to make use of it even if you are not going for Domination, unless you decide you don't want to use it, which is fine, because like Babylon, and Korea, Poland's UA is so strong you can ignore their other uniques completely if you want to.
China, in my mind, has lots of good points, but nothing that stands out as amazing, bar the paper maker giving 2 gold and doing away with maintenance (it's not exactly equal to +3 gold). Everything nudges them toward domination, and they are good at it, and it's not to the detriment of other VC's. Overall, in my mind, that makes them a solid, unspectacular civ.
It's not exactly equal to +3 gold, but if you're going to build a library in every city on your way to getting the National College before turn 100 at the latest, then it does work out to +3 GPT over a civ who does the same thing but builds libraries instead of paper makers. A penny saved is a penny earned. This building works beautifully for going wide which works very well with Crossbow rush, let alone Chu-Ko-Nus. Usually, the more cities you have, the less value each library has, but not so with China.
The Huns are better at dominating small enough Pangaeas, the Mongols are better at dominating Pangaeas in general, and England is usually better at dominating any non-Pangaea, but other than that, China IS the best civ for conquest.
I think the big problem people have with the longhouse is not realizing how good raw is against percent increases.
You only need 1 forest being worked for every 10 hammers otherwise to make it even, which often isn't hard, and forests become good tiles because they give production like mines, but with 1 food. Building a building, with a factory, nuclear plant, and building a building with a windmill (just because that makes it even off, you can use other things), it's 1 forest for every 15 hammers, and 1 forest for every 19 hammers when building spaceship parts with all percent modifying buildings.
Iroquois production is likely to outpace everyone else for most of the game, given their start bias. They'd have to get unlucky not to, and their start bias also means more forest to chop. They need farms, plantations, mines, etc. just like anyone else, after all. Lumbermills don't come too late, either, so the forests you keep aren't sitting there forever, even when they lack deer, fur, or truffles.
The UU isn't amazing, due to being a swordsman. It's good, even great for defending, but it is likely to come a bit later, due to the fact the Iroquois aren't exactly warmongers, though I have had games where I was able to remove someone from the game with a fast upgrade without iron. I do prefer it to Steel (for longswordsmen) though, since Mohawks are able to hold their own pretty well in forests until Rifling, allowing me to put off Steel for a while longer than I might otherwise feel safe doing, and the carryover upgrade is one of the single best in the game.
Infantry upgraded from Mohawks can basically set up impenetrable positions in forests or jungles. Upgraded Minutemen might be better overall, but they don't sit there and laugh at bombers, nor are they strong enough in any situation to be able to put off future military techs, like the Mohawk.
Yeah, I pretty much agree with all of this, especially about the Longhouse, it just takes a little longer to get going than I would like, and my starts have maybe not been so consistently good.
Arabia: Best UU in the game arguably. A chance to get the uber-Pantheon Desert Folklore, and a great UB, and double oil to either sell or upgrade former Camels to tanks eventually. If they don't get DF however they are not nearly as strong.
Brazil: They are amazing once they get going but they do take too long to get going because Jungle starts suck and aggressive neighbours will get aggressive before Brazil gets going. Jungle start is worse for Brazil than for Aztecs because Brazil has no growth advantage and no units that move quickly through jungle except scouts. Brazilwood camps are great but cannot compare to the Aztec UB for catching up to the civs who didn't have to waste worker turns chopping jungle. I really like Brazil but they are pretty much a vanilla civ with a jungle start bias for the first half of the game! If they get a good start though, with lots of production and luxuries and jungle to settle in, sacred path pantheon etc, they will be much better than most civs because their late game abilities are incredibly powerful. In general this civ is weaker on the highest difficulties and in multiplayer.
Sweden: Nobel Prize is OP in single-player where the AI is not programmed to care how many other AIs are giving you +10% GP generation and it synergizes with the other half of the UA. Even if you get no DoF's, 90 influence for a GP gift is a consistent and powerful bonus. As of BNW there are more great people than most civs know what to do with anyway, and you can generate Great Generals just by fighting. Finally, Caroleans are a very nice unit. The downside is the tundra start, which isn't as quite as bad as it's often made out to be. It is intended as a counterbalance to their amazing UA. This civ would be rated higher otherwise.
Austria: Hill start bias is maybe the best in the game. Spending gold to acquire a grown and developed city-state AND its units is very powerful even after this ability was nerfed hard. Coffee House is a pretty good UB and Hussars are a nice little Cavalry replacement if you are going for domination. With three sight and +1 movement they can easily spot for your artillery.
Morocco: Maybe a tier higher than they would otherwise be because of the desert start bias and the potential to get Desert Folklore. If you get Petra in a nice city you will have the best city in the game. If you don't get enough desert to make use of your uniques you can always go and conquer some! The bonus that Berber Cavalry get from fighting in Desert (+50%) is actually bigger than the bonus they get from fighting in your territory (+25%). The bonuses also stack - in your own territory on Desert hexes, they fight as strong as landships! Finally these bonuses are kept on upgrade so conquer a bunch of desert, which probably has oil under it and then upgrade these to landships and they start to get really ridiculous. It's still not as good as Camel Archers though. On maps that have a very large amount of desert Morocco is probably the best civ in the game. In multiplayer games the UA is not as powerful.
Russia: The extra hammers that Russia can get from Horses and Iron early on, the double quantities, and the very large number of extra hammers available starting in the industrial are enough to keep them out of the bottom tier for me anyways, and enough to salvage the tundra start, which is not so bad for going wide. I actually feel it handicaps Russia much less than Sweden because with Russia I have almost no desire to go with Tradition.
Huns: Starting with Animal Husbandry is mostly about getting Horse Archers that much faster. I consider Horse Archers a top 5 UU. The Huns' uniques synergize very well. On some maps they are by far the best civ, on others they are fairly weak.
Ottomans: Still a little underrated. Prize Ships + Barbary Corsairs should give you the strongest navy and your uniques ensure you can fight on land as well. Janissaries are one of the better UU's, the best in the warrior line.
Spain: Obviously difficult to rank. I think I rate them higher than most. In most games it is not hard to find and secure a Natural Wonder even if you have to do it with early conquest using CB's. I think I find a Natural Wonder before other civs in the first 50 turns, roughly half of the time. If Spain does not get to work a Natural Wonder with One With Nature pantheon early in the game, they can still find the Natural Wonders and get the gold and the double happiness, and they can always conquer the NW's or steal them with great generals if they really want to. Tercios are a decent UU. If they start next to El Dorado, they pretty much win the game.
Zulus: Make Ikanda, beeline civil service and then spam Impis. Add siege units. If Pangaea, you probably win easily. This is one of the better civs for domination.
Germany: Much early production saved by acquiring Barbarian units, and then the Hanze is a nice production boost attached to a building you were going to make anyway that comes early enough to make a difference. Panzers come late but seal the deal when they arrive as tank spam is legit as of the most recent patch.
Byzantium: Weak on Deity where it's quite hard to get a religion at all, nevermind a good one, this civ is not as bad in multiplayer or on lower difficulties. As long as you do actually found a religion, your UA will ensure that it is at least a decent one, But, depending on the over civs in the game, you may still find yourself losing out on the best religious beliefs. You'll never beat Ethiopia or the Celts or the Maya, or even Rome if they're trying, to a religion. Both Byzantine UU's are underrated, especially the Dromon. Upgrading units that already have Logistics and Range into Galleasses and Frigates is pretty awesome. Both units are primarily for early defense while staying on the top half of the tech tree but they can be used offensively.
Shoshone: Top tier civ because of the strong start which snowballs. Strong start comes from the pathfinder as well as the UA, which grabs tiles for you and therefore lets you work better tiles earlier in addition to securing them for your empire. This works out to many extra early hammers, apples and gold, and the defensive bonus means you don't have to invest as much in defense as you otherwise would. My favourite thing about them however is that they are the only civ to start the game with a scout, and scouts are what pretty much everyone builds first of all. Also an early CB puts you in complete control of your neighbourhood.