Sure I'll bite, and I'll even include a suggestion for one of the unique abilities
1. England (led by Elizabeth I): Magna Carta: Tile improvements of the same type that are arranged in a triangular position (each touching the other two) get +1 yield from adjacency and cannot be pillaged.
(uniques: Ship of the Line, Yeoman, Royal Navy Dockyard, Redcoat, Market Cross)
2. France (led by Charlemagne): Grand Elan: Combat units receive +2 Strength for every promotion they have earned, and can be upgraded with Faith.
(Uniques: Paladin, Culverlin, Château, Salon, Foreign Legion)
3. Germany (led by Frederick II): Clausewicz Doctrine: Each time one of your military units destroys an enemy unit, receive Science equal to 1/4th of the destroyed units production cost.
(Uniques: Landsknecht, Panzer, Teutonic Knight, Domkirche, Rathaus)
4. The Rus' (led by Olga of Kyiv): Wrath of the Faithful: Your military units do +25% more damage against units belonging to players without a majority religion and barbarians. Clearing a Barbarian Camp grants +25 Faith. Razing a city with a different majority religion than your own grants +50 Faith. (uniques: Cossack, Lavra, Pogost, Strelets, Krepost)
5. Greece (led by Alcibiades): Paedeia: Cities receive +1 Knowledge for every 4 population living in the city. Military units receive +1 XP for each active specialist in the city upon completion.
(uniques: Hoplite, Lycaeum, Odeon, Agora, Pentekonter)
6. Rome (led by Livia Augusta): Marian Reforms: The maximum amount of military units that can be stacked on a single tile is increased by +1. (assumes 1UPT is gone: otherwise, allows you to add another unit to a Corps or Army)
(Uniques: Legionnary, Ballista, Basilica, Latifundium, Dromon)
7. The Abbasids (led by Harun al-Rashid): Proof of God: Double tourism and State religion pressure in all Holy Cities that you control.
(Uniques: Mamluk, Ansar, Madrasah, Noria, Bazaar)
8. Assyria (led by Ashurbanipal): Service to the Šarru: Upon conquering any enemy city, distribute the population that would otherwise be lost across your other cities (those with spare housing have priority). Captured Great Works provide +1 Knowledge and +1 Amenity in addition to their normal yields.
(Uniques: Qurubuti, Hupshu Militia, Siege Tower, Dunnu, Karum)
9. The Hittites (led by Puduhepa): Edict of Telipinu: Whenever you sign a peace treaty with an enemy, sign an alliance with them. Other players cannot break their alliances with you and must honour the aliiance. (but you can break the alliance, at no relationship penalty.)
(Uniques: Mesedi, Bloomery, Mine Factory, Storm Shrine, Golden Spearman)
10. Carthage (led by Hamilcar Barca): Mantle of Tanit: The first maritime building (Lighthouse, Harbour, Drydock, etc) you build in any city is completed instantly. Great Generals and Admirals grant +1 movement to all adjacent units at the start of the turn.
(Uniques: Numidian Cavalry, Quinquereme, Cothon, Tophet, Sacred Band)
11. Egypt (led by Hatshepsut): Valley of Kings: Each world wonder you complete in a city reduces the cost of the next one you build there by 5% (resets to +0% every 5 Wonders). Your Archaeologists can dig up an additional artifact from Dig Sites, and Artefacts of Egyptian culture provide double yields to the owner.
(Uniques: Maryannu, Medjay, Mastaba, Necropolis, War Barge)
12. Ethiopia (led by Taytu Betul): Eternal Flower: Each time you settle a new city, grant it +5 Production and +5 Faith per turn for 50 turns. (reduces by -1 every 10 turns until you hit 0 on the fiftieth turn.)
(Uniques: Beit, Hawult, Mehal Sefari, Oromo Cavalry, Shotel)
13. Persia: (led by Mithirdates I): Kvarenah: When declaring Friendship, automatically enter a Research Agreement with the other Civilization. Each consequetive Research Agreement signed with another player increases the Science boost by +5% for both.
(Uniques: Immortal, Pairidaza, Zradha Shirvatir, Satrapy, Farhangestan)
14. Mongolia (led by Mandukhai): Qural-Dai: Barbarians are friendly to you, while City States are hostile. Defeated Barbarian units have a chance to join (33%)), double if they're Cavalry. Clearing a Barbarian Camp grants an additional reward: (Choose between: A builder, A random military unit or a city on the spot of the camp.)
(Uniques: Keshik, Ger, Yurt, Mangudai, Huihui Pao)
15. China (led by Yongle): Feng Shui: Tile adjacency is decided by variation on the surrounding tiles. Each unique improvement (including World Wonders), feature (including Woods, Mountains, Rainforests, etc) and District type grants +1. Each recurring improvement, feature or District type grants -1. Featureless unimproved tiles are considered neutral. Cities with perfect Feng Shui are always Euphoric, regardless of amenities; Cities with a Feng Shui rating for 70% or higher are always Happy, regardless of Amenities.
(Uniques: Zen Garden, Theatre Pavillion, Chu-Ko-Nu, Fire Lance, War Junk)
16. Japan (led by Masako Hojo): Bushido: After receiving their third Promotion, All military units upgrade into Samurai, unlocking unique promotions and abilities that are otherwise unobtainable (cf: Samurai are a unit class on this occasion, rather than a unique unit)
(Uniques: Dojo, Shoen, Akatebune, Ashigaru, Torii)
17. India (led by Ashoka): Beloved of the Gods: Every city can support two Pantheons instead of one - one is picked when the city is settled and applies only to that city, the other when Maurya earns their overaching Pantheon, which applies to all cities. The same Pantheons can be chosen multiple times, and their effects stack.
(Uniques: Varu, Sepoy, Stepwell, Stambha, Sala)
18. The Inca (led by Topa Inca Yupanqui): Quipu: At the start of your turn, gain a 5% interest on your unspent Gold and Faith.
(Uniques: Kamayuk, Waraq'aq, Qhatuq, Andén, Pukara)
19. The Lakota (led by Sitting Bull): Ghost Dance: Your military units survive their first killing blow with one HP left.
(Uniques: Teton Rider, Heyoka, Inipi, Buffalo Pound, Sundance Circle)
20. America (led by Thomas Jefferson): The American Dream: Bonuses from positive happiness to city yields are double in your empire. Your Happy Cities provide +1 additional World Congress vote, while Extactic Cities provide +3.
(Uniques: Pioneer, Minuteman, Sherman Tank, Homestead, Football Stadium)
Feel free to C+P this, Firaxis.