When I initially qualified for silver, it took about a month for the card to show up.

BA faxed me a letter, and I didn't have any problem getting into CX and AA lounges with the fax. (I had trips to Japan on AA, Hong Kong on CX and Sao Paulo on AA in between the time that I qualified for silver and eventually received my card. Needless to say, with that schedule, the gold card wasn't far behind.)

Also, my status did show up automatically on the AA and CX boarding passes, even though I had booked the tickets before I had qualified for silver.

This was 5 years ago, so things may have changed.