Hodgkins-Byrne, 29, and Wilde, 26, made a strong start to their final but were overhauled by eventual champions New Zealand and silver medallists Romania.
They had to dig deep to hold off the Netherlands to secure their medal.
Hodgkins-Byrne, who carried on training while pregnant, did not want her rowing career to finish after the disappointment of Rio, where she came seventh in the quadruple sculls.
And, like Glover, she has also spoken about how motherhood has given her a better perspective on life.
Asked what her medal proved, Hodgkins-Byrne said it was “completely possible” to balance being a mother with taking part in elite sport.
“I’m so lucky with Becky and my coach Tom (Pattichis),” she said. “Whenever I’ve had an issue with childcare or sleep or anything, both of them have been so flexible and understanding. It’s never become an issue.
“I’ve been able to be Mathilda the athlete rather than Mathilda the mum trying to juggle everything. I’m so grateful for that.”
Wilde, who swam for Wales before taking up rowing, said: “It means everything. It was a dream come true to make it here, and to win a medal is beyond anything we could have imagined.”