Ok, so I got the spare part, but I can't seem to be able to replace the one that's not working on the chopper. Everytime I try to interact with the chopper Cord says "We get the spare part and we get back in the saddle", nothing more.
I can't figure out if it's a bug or I'm missing something...

Have you rescued Alex yet?
You should have got a limpet mine where you got the spare part and then you should blow up Alex's cell door. After talking with her you should be able to replace the damaged part.

Yes, I already rescued her and she's waiting next to the chopper.