England will face France in the bronze-medal match on Saturday (22:00 BST) – and victory will mean this is their best performance in a men’s World Cup since 1966.
“It is heartbreaking to be so close,” said Bullingham.
“The players and Thomas gave it everything today and the squad, coaches and staff could not have worked harder during the tournament.
“I would like to thank them all – and also give my heartfelt thanks to our wonderful fans here in the USA and at home.”
Despite the disappointment of losing in the semi-final, England’s progress will be viewed internally as a relative success.
Tuchel said at his post-match news conference in Atlanta: “We keep on going with the contract until the home Euros.
“I’m looking forward to that even though right now it’s difficult to look that far ahead.
“Of course, it’s a semi-final; a lot of big football nations are eliminated before the semi-final, so it is an achievement.
“No-one wants to hear that at the moment; me neither because we demand the most of ourselves. That’s just the nature of being competitive.”
