Part of that idea for conversing without 'uh/uhm' came from employee-to-employee advice I read somewhere about preparing memos for meetings with Jeff Bezos at Amazon. It was something along the lines of: have all your facts and answers ready but don't put everything in the memo; economize the information you put to paper down to what's essential, so he doesn't waste time reading more than he needs to and has questions to ask, but make sure you have the answers ready.
That same advice can be adapted to conversation. Figure out how to communicate one discrete idea, but don't try to answer every concern/question the listener might have before they ask about it, even if you have those answers ready or are thinking about them as you talk. Let the other person speak and engage with the idea, then answer the questions they do ask. If you come to an 'uh/uhm' point in your own words, the other person probably has a question/concern/thought already (and it might not be the one you're about to 'uhm' into).