Having the same problem...I solved reading the doc here:
https://developer.android.com/reference/android/webkit/CookieSyncManager
For future readers: to force the Cookie sync process you can manually call the flush() method of CookieManager
I personally put in webview the following code:
public class MyWebViewClient extends WebViewClient {
public void onPageFinished(WebView view, String url) {
CookieManager.getInstance().setAcceptCookie(true);
CookieManager.getInstance().acceptCookie();
CookieManager.getInstance().flush();
}
}