British Airways stops sending physical cards to Executive Club members
Instead, members should use the app or the BA website to view their membership details. Most of the time, a Oneworld airline-issued boarding pass should already display the member's status level and thus grant access benefits, such as lounge access and priority boarding. However, if it doesn't, the BA app has an electronic membership card under 'My Account', which can be used should proof of membership level be required to gain access to lounges or other status perks.
Generally, we recommend adding frequent flyer details to any booking either at time of booking or before traveling so that boarding passes and the airline's system correctly displays status and any entitled perks. In addition, it also ensures that travelers earn both Avios and Tier Points for any flights taken by Oneworld alliance carriers.
The need to physically produce a membership card to prove status has greatly diminished over the years as airlines have invested in their IT systems and app capability.
BA has confirmed to TPG that any members who would wish to receive a physical membership pack with a membership card can order one through their Executive Club account on ba.com.