Let me also greet you to Flyertalk and welcome you to the BA forum
michieluc. It's great to see you here, I hope you continue to participate. The incentive for that is that if you keep an eye on this forum you should be able to fly further forward up the aircraft for less money, at least in theory. Many FTers have this experience!
That fare seems high for one person, the cheapest NYC fare is currently around £368 . Fares will be slightly cheaper tomorrow, due to the rise in Sterling, since the US taxes will become cheaper in Sterling terms, but that's only a small amount. In this post, the fare has gone down by 40 pence over the last 5 days purely due to that.
https://www.flyertalk.com/forum/29628570-post104.html
The rest of the thread may also give you some insights into the current fare wars too, click on Thread link on the top right. As you can see that seems a high fare if one person. By all means give the exact dates, and limitations on your travel dates, if you want an expert to see if they can find a cheaper option.
As a general point, if you see a fare which is good value then nab it, and stop looking after 23 hours. Why? Because after 24 hours there are likely to be large change fees, but none before. Trying to second guess airline fares is something that even experts here can't easily do, so if it is affordable, buy it, then stop looking and start planning.
Personally I think that if travelling in economy then by definition you want the best fare. So unless you are very tall or something I would save your money and use it more enjoyably in the States, it's a fairly short flight and you'll be there very quickly. That's just my personal view. But yes, BA charges for seats, and if you are minded to do that you best do it at the point of purchase, the cost goes up if you do it later.