My understanding of the scenario you are trying to implement is that you have the following actors:
Browser -> Web Application -> Web API
Where the browser is authenticates a user to the Web Application and then the Web Application attempts to obtain a token that will allow it to access a resource at the Web API on behalf of the user.
That being the case, your Web Application will need to redirect the browser to the AAD OAuth 2.0 authorization endpoint. There the user will enter their credentials and be redirected back to the Web Application with an authorization code. You can then use that authorization code to obtain a token via the Active Directory Authentication Library (ADAL). Specifically you will use the AuthenticationContext.AcquireTokenByAuthorizationCode method. The following blog entry by Vittorio gives more detail on the code you will need to implement in the Web Application in order to obtain the token. It is not an OWIN specific implementation, but should be easy to translate into your NancyFx app.
Using ADAL’s AcquireTokenByAuthorizationCode to Call a Web API From a Web App
You should not have to change your ASP.NET Web API that you implemented per "Protecting a Self-Hosted API with Microsoft.Owin.Security.ActiveDirectory"
The version of AuthenticationContext.AcquireToken that is used in the client app example in the Protecting a Hosted API... blog entry is intended for a different scenario where no browser is involved. Instead the user is interacting with a desktop application that is then calling a Web API on the users behalf. In that case, the AcquireToken call must show a browser based dialog in order to allow the user to sign in and obtain an appropriate token.