I know a couple of people having problems accessing some HTTPS web pages via their BT Business connections - both using BT Business hubs on ADSL2+.
Searching on Twitter I can see this (viewing secure HTTPS pages) seems to be something that pops up quite regularly over the years for BT customers!?
Unfortunately the Twitter conversations don't really go anywhere so don't help.
Rebooting the router seems to get things going for a while before the problem returns the next day. Their exchange is currently mentioned in the generic "broadband issues" outstanding faults on BT's status page - due to be cleared on the 21st. But it doesn't say what issues they are fixing just:
"A small number of our customers in the areas shown below, may experience a loss of telephone and/or broadband services. We hope to have service restored as quickly as possible and apologise for any inconvenience this may be causing. Services can be affected by a variety of reasons such as damage caused by 3rd parties or cable theft."
When/if I get to visit I can investigate further but thought I would ask here in case other BT customers have been there, done that...