How do i make logins happen via Amazon Cognito REST APIs (for user pools) on platforms for which there is no official SDK? - Note that i am asking
From what you have discussed, I consider you are trying to do that from a web frontend. Cause, cognito is providing you the necessary backend support and it expects you to communicate(e.g. authenticate, sign up etc.) from a presentation layer- that's why you found SDK's for different mobile platforms. They also have SDK for web app- the access is available via their Javascript SDK.
Here's a detailed tutorial to achieve what you have asked from a web frontend using their JS SDK- Accessing Your User Pools using the Amazon Cognito Identity SDK for JavaScript