From what I understand from the ICA website, if you go through their online portal, it will take 3 working days for them to process your application upon receiving all the relevant document. If you are going down in person to do it over the counter, you will be given a receipt and be notified when your IC is ready for collection. If I am not mistaken, your name change will be recognized once it is on your updated IC. You can also refer to this post written by someone who has gone through the process. Some helpful tips there on what to follow up with: https://innersanctuary.wordpress.com/2012/06/27/changing-your-name-legally-in-singapore-the-how-to-step-by-step-guide/
From what I understand it takes 28 days for it to be legal.