Competitive solution with Air India is available on route London - New York. If you are flexible to book your ticket a bit in advance and you want to travel to NYC in first quarter of 2017 you can book direct flights from UK to USA already for £316. All charges and taxes as well as checked luggage are included in base fare. 
To book this deal with Air India to New York you need to look for travel dates between January till end of March 2017. Valid for bookings made till 30th of September 2016 or until sold out. Below you may refer to booking and travel date samples.
Booking sample
Travel date samples
6. - 13.1.
13. - 27.1.
30.1. - 10.2.
17. - 24.2.
10. - 25.3.
31.3. - 15.4.
London is connected with rest of the Europe by many low-cost carriers. The easiest way how to book your connection flight is through Skyscanner. To book your accommodation you may consider promotion codes offered by Hotels.com, Ebookers, Venere or Hotelopia.
Booking sample of cheap direct flights from London to New York from £316 here. In this case you would depart from UK to America on 24th of February 2017. Your return flight from USA back to Europe would be on 11th of March 2017.
