Have you tried?
mWebView.setDownloadListener(new DownloadListener() {
public void onDownloadStart(String url, String userAgent,
String contentDisposition, String mimetype,
long contentLength) {
Intent i = new Intent(Intent.ACTION_VIEW);
i.setData(Uri.parse(url));
startActivity(i);
}
});
Example Link: Webview File Download – Thanks @c49