Britain has a history of not trusting its politicians, so this has never been popular, while America loves its politicians until they do something wrong. We name our children after family and royalty, and I doubt this will change at all soon.
I agree that Winston has the potential of catching on, but I doubt it'll be because of Churchill.
It's a different dynamic there -- it's not the politicians that are loved, or even know much. It's to identify as 'American', as opposed to expressing any kind of admiration for them as individuals.
History is taught in the US as a march of progress towards liberty and happiness, with Anglo Saxon/Scottish immigrants filling in the vast country. This sort of history teaching was especially in vogue in the early 20th century, and as more and more people arrived without a word of English, to face suspicion and hostility, they gave their children what they saw as quintessentially American names that they gleaned from rudimentary readings on American history.
Victorian Brits loved naming their children after politicians -- especially using their surname/firstname-and-surname as middle names. "John William Ewart Gladstone Baker" is one example. I wrote an article about the topic: here
I agree with previous posters that their is a different cultural dynamic and emphasis on education now. Politicians are no longer the heroic leaders of an empire, forging the industrial future of the country. It's unlikely that parents would deliberately name their children after living PMs, but some might be tempted to use long-dead PMs as namesakes if their legacy has passed favourably into posterity.