You book to fly with Virgin Atlantic and you fly with Delta Airlines.
When you book you are given excellent flight times to cope with the horrendous queues in New York. Shortly after receiving your travel details your flight times are drastically shortened giving you no time to queue, go through security and collect your cases before heading to your next flight. If you miss your flight you are advised that you are not entitled to another flight as Virgin and Delta are not responsible for the queues in JFK.
There are no staff to assist you and no announcements are made. You are basically on your own. My wife and I were really disappointed with the customer care as there was none. Try another airline that will give you good customer care, not abandon you at the airport. I was offered vouchers when I wrote to Virgin, but as my wife and I would not fly with Virgin again, I requested that they kept the vouchers.