You can't usually see much details, merely a list of cases and more recently there are automated emails after submission so you need to make your own number record. The way to see if cases are open (but not the details) is to login to BAEC then in the same window type
ba.com/complaints
and that's as good as it gets. Unfortunately, and separate to the above, the Customer Relations team got badly burned by moving their system to Salesforce and a huge backlog developed, now stretching past 6 months for some cases. Many complaints are dealt with much more rapidly, if they are straightforward and short (2 sentences) but anything complex is now taking many months to be resolved, 2 or 3 months is quite normal for Gold card holders. Also, there is a bot that closes automatically duplicated - or suspected duplicated - cases. You can call up or use social media teams to check status / processing, but they won't actually resolve the issue, merely confirm that a case is open and will be progressed in the next 6 months. You do have the option of CEDR or MCOL at 8 weeks (or before that with MCOL).