You convert twice, and get the bonus twice, if both your BAEC and avios.com accounts meet the criteria.
Not sure what the BAEC criteria are, but for avios.com the account must have been open at 1 Dec 2011, you must have collected 1 Avios between 1/12/10 and 1/12/11 and you must have done NO Tesco conversions through the account.