Has the accent between native English speakers decreased in recent decades with the advent of TV and now with the internet?
In the 20th century there was a push primarily among actors, broadcasters, and aristocrats to use a non-vernacular accent called the Transatlantic accent, which was a blending on American and British accents that was meant to be more universally understandable than any native accent. It was also meant to show off one's education because the only people who spoke with that accent were trained to through high class education.
However that accent did not become standard or commonplace despite increased connectivity between different regional accents of English. Perhaps a natural and more gradual progression towards a shared accent is possible, but an explicit attempt to make that happen did not take hold. This is at least some evidence against the idea that we would shift towards a more shared accent among international English speakers.
This is really a question that should be answered by research. I did a quick Google search, and came across this article.
In the abstract, they say:
Our analysis suggests that although there has been an obvious reduction in regional variation with the loss of traditional dialects of English and Scots, there has not been any significant convergence (or divergence) of regional accents of English in recent decades, despite the rapid spread of a number of features such as TH-fronting.
If I understand this correctly, although some traditional dialects may have died out, you'd still expect to hear distinct British, American, Australian, etc accents. This article was posted in a journal called "Language Variation and Change", which might have other interesting studies that can help answer your question.