In reply to Andy Johnson and jimmccall
"You can log in or register at https://mso.thebmc.co.uk"
That's a very helpful post for others to see as it explains where you can see you card (You can't download it but at least you can take a screen shot and print it.) Unfortunately there's no link to the mso site from the main BMC site
However, since my original post I have had some success in my research
On 7th March 25 the BMC sent out one of their usual 'interesting' emails, the first few pages of which were devoted to discussing budget matters and some new member of the board. If you persevered in reading then at the bottom was a small note saying that Summit would become digital soon and that so would the membership cards, but no further details.
Then on 29th May (two months after club's 2024 BMC membership cards ran out) came another email showing the link you gave to https://mso.thebmc.co.uk. However the actual url behind the visible link wasn't the same so my antivirus software assumed it was either spam or malicious and quarantined it. As far as I can tell there have been no further reminders or specific notices, either by email or on the web site.
As it happens I use the mso page to administer club membership for all the members of my club and logging on just now I was offered the chance to link my personal membership with my admin account for the club. Lo and behold my membership card appeared!
Regarding reciprocal rights, maybe I use the wrong words. For the last forty or so years I have always been given a big discount in CAF huts, (often in the Ecrin!) something like 50%, just by showing my BMC card. That's what I meant. Whether this is true RR or something else I don't know. Using the mso site it recommends adding your card to Google Wallet but I'm not sure a QR code on my phone will work as well as a physical card in a hut. I'll print out a screen shot of the original looking card from my MSO profile page! Thanks to all who responded.