Some years ago I encountered that issue In the crossover period like during the days / weeks after end of month when the “demoted” SG physical card hadn’t arrived and I was off to the USA
From memory
The physical BP said bad luck demoted while the WP card you’re still in.
At LAX AA gave me the benefit of the doubt (bluff your way in nicely)
On the return trip they didn’t as that was some weeks later.
PS I NOTE back then at JFK there was one entrance for everyone flying QF to LAX and you were directed right to the J lounge
one could turn for example turn left to the WP Lounge 
and just plead “didn’t know which way to go”
