Technically speaking, I'm a big fan of http://swagger.io/, especially since it's turned in to the Open API Initiative. You (or your client) define an API in YAML, and then you can use the Swagger Codegen tool to automate building a client SDK and a mock server based on it. API changes are handled by simply rerunning the generator. I've found that the NodeJS/Angular code that's generated is good enough to use in production if you're sensible.

As far as managing the process, the only way to do it is to make sure the customer understands that the lack of a working API test server (with good stubs to start with, and actual data further on in the project) will delay the project. Make sure that you tell them the app development will lag behind the API by at least a month.