Greenwood – Tindall's partner in midfield – flew back to the UK after scoring England's try in the 25-6 win against South Africa to be with his wife.
The couple's first child, Freddie, died after being born prematurely last year – and Greenwood chose to keep the present situation to himself until after the Springbok game.
"We don't know very much about it all, but for Will to play the way he did was outstanding," said Tindall.
"All our hearts are with him and Caro. She's out of intensive care now, which is awesome.
"Will is captain of the attack. He's a great thinker, that's his main strength – the way he reads the game. But the other boys wouldn't be here if they weren't good enough, everyone knows what we're trying to achieve."
Asides from Greenwood and Bath lock Danny Grewcock – who is nursing a broken toe – England have a clean bill of health, and all other 28 players took part in training at Scotch College in Fitzroy on Tuesday, ahead of the game against Samoa on Sunday.
The management and players have analysed their performance in Perth and highlighted the areas to improve after defending for long spells against the Springboks.
Tindall is confident the team will continue to improve throughout the tournament if they can pick up the tempo in attack.
He insists that they are aiming to emulate Australia, who conceded just one try in the whole tournament before lifting the Webb Ellis Cup in 1999.
"I thought we defended very well, but we didn't attack too well against South Africa. We seemed to lose a bit of rhythm," said the centre.
"We have to make sure we go into the game against Samoa with more cohesion. But one of the things we said before the tournament was that the World Cup is normally won by sides with the best defence. Nobody has scored a try against us yet."