Street and the England squad will be back in London from Paris on Monday, having secured the World Cup following a 21-9 victory over Canada.
England had lost the three previous finals - all to New Zealand - with Street's starting line-up against Canada featuring 11 survivors from the team beaten by the Black Ferns in London four years ago.
"It feels a bit like a dream," Street said.
"It's quite surreal, and I think I will wake up in a minute!
"It's a pretty amazing feeling, and I am just so proud of everyone involved. It's a really special group. We've been on a long journey, but to have that trophy makes it worthwhile.
"The support we've had has been amazing, and it's a huge relief to be able to deliver."
Captain Katy Mclean hailed matchwinner Emily Scarratt as "a phenomenal player" after England triumphed at Stade Jean-Bouin.
England's success was secured by 16 points from centre Scarratt, including a superb late solo touchdown, and a try by full-back Danielle Waterman.
"She's a phenomenal player, world-class, and world-class players do the world-class things. She absolutely delivered for England," Mclean told Sky Sports 4.
"That group of girls and staff deserve everything we've got today.
"This is all about the England rugby family. It's amazing. It started years ago with the people that have brought us here, our family and friends."
Tributes quickly flowed to the England side, whose triumph comes 11 years after the men's team were crowned world champions following a dramatic victory over Australia in Sydney.
Minister for Sport Helen Grant said: "Congratulations to the fantastic England women's rugby team and all their support staff on becoming world champions. They have made the nation proud.
"I am sure they have inspired many women and girls to get involved in the sport, and I know the RFU (Rugby Football Union) will be working hard to build on this success."
And England 2003 World Cup heroes Lawrence Dallaglio and Richard Hill were among many past and present international stars to praise the achievement of Mclean, Scarratt and company on Twitter.
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