How much time do scikit classifiers take to classify?
I don't see a huge problem here. So, I would try to answer all of your questions from the production-level point of view: 


  How do I confirm that classification will not take much time?


Take a subset of the corpus data you have (you can randomly do it, no need of sampling), and test your algorithm on it, and they approximate/generalize it to the overall dataset.

(SVM is comparatively faster. Nevertheless, do the above process just to be sure.)

And do test it in the development environment before pushing to production.


  Is using Python's scikit library for websites/web applications
  suitable for this?


Yes, it is. It is already being used by a nice chunk of companies out there.

The third question about Amazon and Flipkart cannot be answered by someone outside their teams.

In addition, I would advise you to use the mapreduce techniques for training your models. And as already advised, pickle your models so that you don't need to train them with every request.