british airways have decided not to inform their customers which airport they will be travelling from (let alone which terminal)

 Multi-choice.  You're an airline.  Should you (a) tell your customers, at every stage there is flight info, their booked flight details including airport and terminal, flight number, etc; or (b) just put vague details like the city, and be ever so squidgy about everything, because you heard about being cool and minimalist.

You're a business.  Should you (a) put all relevant information on the confirmation page from the web site where the purchase was made; or (b) the "confirmation page" can just say you've sent them an email, which may or may not arrive, sooner or later, subject to the whims of email, a completely different medium from the medium of the "confirmation page".

The first question is a trick question.  The question says you can maybe seem cool by not bothering with things like airports and terminals if you're an airline.  But the correct answer is (a).

The second question is a specialisation of not switching from web to email half way thru something, so hopefully everyone got (a).

How do British Airways do on this?

That's their confirmation page.  It has the departure as "London".  That's not quite specific enough.

On the first question, then, they've failed.  The airport is not shown on the confirmation page.  Let alone the terminal.

On the second question, also fail.  The confirmation page does not show relevant information, instead stating they've sent me a confirmation email.  At the time of writing, around half an hour later, this has not arrived.

How easy do they make it to find out the terminal and airport I need to go to?

Logging back in to their web site, because it has a very short auto-logout, one might think that a booked, upcoming flight would feature prominently on the front page.  It's there, but not that prominently.  I have to scroll down for it.  Whoever designed the page thought I'd be more likely to book more flights, than to look up info (that they haven't given me yet) about an already-booked flight.

Clicking on "Manage" brings up a form, where you can search your enormous database of upcoming flights, which in most cases consists of one single flight, by typing in a booking reference and name.  Never mind that you're already logged in, and so it knows who you are.

Anyway, going with "Your next flight", this does contains the airport and terminal (but not the flight number. Why?), so the quest is over.

This is quicker than the way I found first, which involved something like trying links like "print confirmation" and "itinerary" from the booking page until I got one with the airport and terminal.