Abra can be found early on in the route just above Cerulean City. If you don't manage to fine one, however, you can get them in Celadon Game Corner for 180 Coins. For the game corner one, I recommend saving before purchasing it and keep resetting until you find a Timid/Modest one.
While Mr. Mime can be quite offensive, it serves mainly as a Special Wall - which isn't that important in-game unless you plan to play competitively. Kadabra is strong and fast. It easily carries you throughout the Nuzlocke so you should really go for it.
Kadabra @ Any
Ability: Synchronize / Inner Focus
Recommended Nature(s): Modest / Timid
- Calm Mind
- Any coverage move
Hope this helps!