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Unfortunately, if you redeem on BA, you will be hit with hefty fuel surcharges, especially in the premium cabins, when you redeem AA or BA miles. You might want to look into some of BA's other partners (like LAN) to see if they also assess fuel surcharges on BA redemptions between the US and the UK.
If you are only willing to fly BA or VS, and if you cannot find a program that does not assess fuel surcharges on them, then you might want to look into whether the fares you will be paying for your work travel earn full credit in ANA's FFP. Because ANA's redemption rates are distance-based, rather than zone-based, you can get a J roundtrip between NYC and LON on partner carrier VS for only 63,000 ANA miles (plus fuel surcharges).
Check out ANA's website for more earnings and redemption info.
But frankly, you might want to give AA's Business Class to London a try. Many people prefer BA's J seat to AA's J seat, but some prefer AA's J food to BA's J food. And AA also offers F for 62,500 miles each way, compared to J for 50,000 miles each way. You might prefer to spend more miles to fly AA F, than fewer miles and a lot more cash to fly BA J.
Also, check the threads here on MilesBuzz! about credit-card deals with huge bonus-mile sign-up offers, especially the Citi/AA Visa and Amex cards, and the Chase Sapphire card, which earns points that can be converted to BAEC miles.
Good luck!