Okay, everyone take a chill pill.
BA has a rule that they will not hold a plane for anyyone, wants you to be at security no later than 35 minutes before the flight, and be at the gate no later than 20 minutes before departure.
This is true for all originating passengers. However, things change when the passenger is making a connection from a BA flight to a BA flight.
The minimum connection time allotted for an international to international flight in Terminal 5 is 60 minutes. This is not the airline rules but the time agreed upon with the airport. It is felt you need a minimum of 60 minutes to make the connection.
The flight into LHR was 15 minutes late. You now have a connection time of 45 minutes--below the 60 minute window. If using the above rules, you'd have 10 minutes to deplane, walk and then take the train to flight connections, and then on to security. Not going to happen. You would then have 15 minutes to get through security and get to your gate. Also probably not going to happen.
BA knows you're late. They know your incoming plane was late. It's possible there were others on that Rome flight going to Seattle or passengers from other delayed flights. However, in flight operations for BA a decision was made to hold the plane. This is not a light decision as new take off slots have to be requested. (Unike in the U.S. where flights are first come, first service, in Europe, airlines have to request a take off slot. If they miss theirs, they have to request a new one. ) A short departure delay can usually be made up in the air and arrival times wouldn't be affected. It's not the on time departure that the airlines care about it's the on time arrivals. And there is usally enough wiggle room on long interational flights for this.
Here's where it gets tricky. By arriving late, you were not allotted that 60 minute window. So, if the SEA flight left without you, and BA couldn't get you to your destination within a certain period, you could be owed compensation--not forced to buy another ticket. It has nothing to do with jars of honey or train inspections. What if neither happened and there were twice as many people going through security? By delaying the flight, and giving you the full 60 minutes for the transfer, BA would no longer be liable to give you compensation. They would put you on another flight but owe you nothing.
The delay would not have shown up on the departure board so other BA employees would not know unless they went into the system and looked at the flight. (This would have wasted time.)
So what have we learned:
1) Airline rules are just that. Rules, not laws. They can be broken. But not by you.
2) In the future, if concerned, try not to book tight connections as that would leave you some breathing room
3) If you think you might need assistance getting around the terminal or from one gate to another, let the airline know in advance so it can be arranged. Requesting help after you arrive will take a long time to get.