I just did this with the chase continental card... i don't see why it wouldn't work to get you the miles quicker.
But it's already rather late to get award tickets booked for july/august which is prime time for europe. I have a lot more experience with united/star alliance and to do a mid summer trip w/ a saver award, you often have to book 6-12 month in advance. I just did a search for sea-lhr through BA.com and there is nothing in economy and scattered days in August/September for premium economy and business. You don't need the miles to search for award tickets via your BA executive club account.