Data point - I applied for the United Club visa around beginning of April this year. Received the physical Club membership card around middle of May
I received an email from United News & Deals <
[email protected]> with the subject "
How your UnitedClub membership works" on Apr 13, and within the contents of the email, it says "We’ll also send a physical membership card in the next eight weeks for you to use when visiting Star Alliance™ partner lounges."