We have just used Avios to book our flights for next year. Economy flights from Heathrow to JFK (out with AA, return on BA) cost us £350pp in taxes and supplements on top of the 40,000 Avios miles. In comparison, our onward Business class flights from JFK to Los Angeles with AA cost us 25,000 Avios each way but £1.68 (not a misprint!!) in taxes, the flight being around an hour less than the Transatlantic leg.
One tip I would give you is to check both the BA Exec Club and Avios websites for flights. Allegedly they use the same allocation of seats, but for the date of our return flight from JFK to Heathrow, the Avios website is showing availability on all BA fligts, and with a good choice of Ecomomy, PE, Business and First. In comparison, the BA website is showing only two economy seats on the daytime flight and that's all
One advantage the BA website does have is that if no BA flights are available, you can then search for availability on partner airline (AA or Iberia) - this is how we got AA flights outbound, but as above, used the Avios website to book the return flight with BA.
Also an important point is that BA and Avios start releasing flights up to 350 days in advance, so keep checking on a daily basis for availability.
Good luck with getting a flight sorted.