I'm not sure BA is the best way to go... for 1 big reason - The fuel surcharge is outrageous (at least compared to what domestic airlines generally charge for USA-based trips).

I just priced out a JFK-LHR return flight (low season) and it cost 26,000 miles + $693.22 in surcharges.

From NYC, if you're willing to take connections, I'm sure you can find cash fares not far from $700...

Flying NYC-LON, and within Europe, Delta miles can get you to many places as well. However, credit card signup bonus and transfer partner is not as easy to come by.

Since you're applying for credit cards, the United Air Chase card, and the American Air Citi cards both have decent sign up bonuses... They are useful for getting you across the ocean cheaper than with BA miles.

To be blunt, NYC-LON is one of the easiest/cheapest transcon flight, with lots of competitors to keep prices low. You should consider using your hard-earned miles on something more valuable.