The following Pokémon become available after obtaining the national Pokedex in LeafGreen:
Sentret and Furret
Ledyba and Ledian
Spinarak and Ariados
Crobat (via evolution)
Pichu (via breeding)
Cleffa (via breeding)
Igglybuff (via breeding)
Togepi and Togetic (via an egg given by an NPC)
Natu and Xatu
Marill and Azumarill
Politoed (via trade evolution)
Hoppip, Skiploom and Jumpluff
Yanma
Slowking (via trade evolution)
Misdreavus
Unown (all forms)
Wobbuffet
Dunsparce
Steelix (via trade evolution)
Heracross
Sneasel
Slugma and Magcargo
Swinub and Piloswine
Remoraid and Octillery
Mantine
Kingdra (via trade evolution)
Phanpy and Donphan
Porygon2 (via trade evolution)
Tyrogue (via breeding) and Hitmontop (via breeding then evolution)
Smoochum (via breeding)
Magby (via breeding)
Blissey (via evolution)
One of Raikou, Entei or Suincune, depending on your starter (roaming)
Larvitar, Pupitar, and Tyranitar
Azurill (via breeding)
Wynaut (via breeding)
Note: Espeon and Umbreon are always unobtainable in FR/LG, even after obtaining the national Pokedex, due to the removal of the in-game clock.
Note 2: this is for LeafGreen only. Version-exclusive Pokémon for FR/LG can be found here.
Source: looked through my own copy of LeafGreen, and consulted this to make sure I wasn't missing any trade evolutions (since I have no way to trade with this game).