I hope that I understand correctly your use case.
If you want to use oAuth, then your mobile apps are considered as the oAuth-client.
Your "server" holds the "protected resources", and it can be used only with oAuth access-token, so it is called "resource server". Now you want something to supply this access-token, so this is the identity-provider, AKA authentication server, e.g. Facebook, Google, (or implement one by your own).
The flow is (generally): the user (mobile app) tries to reach a protected resource; since it has no token, he is being redirected to the auth-server. the latter is responsible for the user/password login page, and creating the token.
If it is true - you still can implement everything by your own, without using Facebook/Google APIs, because oAuth has SPECs. However, it can be easier for you to use the providers' packages.
EDIT: reconsider the usage of oAuth
You use oAuth only if you want your webapp to support oAuth SPEC. There are several benefits, one of them is that you can use 3rd party identity provider, e.g. Yahoo! and use their identities without managing them. So if I have a user in Yahoo!, I can use your app without additional registrations (your app will have to support access-tokens from Yahoo!). But in your case, you are about to implement all the logic of identity-provider (forgot password, change password, registration, etc) plus supporting oAuth - and all of this without enjoying the benefits of oAuth at all! So - you have to reconsider the usage of oAuth...
resource server
andauthorization server
will be same (Joomla in my case). I don't see why I would use Facebook or Google APIs as I don't plan to have users log into their mobile apps with FB or Google? They need to authenticate with Joomla, so wouldn't I have to build my own implementation? – Arcadia