I *think* I was told that if the airline causes the delay then they update the EOBT. If ATC cause the delay ONCE THE AIRCRAFT HAS REPORTED READY IN ACCORDANCE WITH ITS EOBT then its up to ATC to input the revision to the EOBT.
In practice this rarely happens as LHR did have some dispensation when the rules came into effect and nothing has really been done to ensure we properly apply them. Once big delays are in force then EOBT updates I suspect will be lower down the priority list. In previous control towers ATSAs did a fine job of keeping EOBTs up to date if required, however times have changed and those in charge thought that not to be the best way to deal with this and other issues of ATM......