IAH has been announcing these Terminal D to E check-in moves airline-by-airline on their social media channels over the past few weeks, typically the day before each move, but nothing yet for BA... worth keeping an eye on their X/Instagram/Facebook feeds for the latest.

TSA in D is still closed, so if you’re checking a bag I believe you will still need to start in D, then head to another terminal for security. HBO with boarding pass can head directly to any terminal with an open TSA checkpoint.

Edit to add: the airport and airline websites have long shown D for BA departures and E for BA arrivals. BA flights mainly arrive and depart from gate D3, but all international arrivals go through the central immigration and customs processor in Terminal E.