It may depend on what you mean by "gender roles".
Genders seem to be less stereotyped than they once were ...
... but the genders still seem clearly separate from each other.
Every person they meet is clearly described as either male or female: the man in the museum, the little boy expelled from school, his mother whom they call a lady. It's still easy for the children to tell men and women apart from each other.
What's more, the clothes they wear aren't described as being androgynous. Given all the other things in the future London which are remarked upon as being very strange and unfamiliar, I expect it would have been at least mentioned if the boy was wearing clothes more associated with girls. As Peter Shor notes in a comment, young boys wearing dresses wasn't unheard of in the time the book was written and set.
Your quote about "lady" being a "quaint old word" is interesting, but remember that the word "lady" has more connotations than "woman". It suggests being ladylike, possibly even noble or of high social standing. In this utopian future, class boundaries seem less pronounced or even non-existent:
'But does everyone have rooms like this, poor people and all?' asked Anthea.
'There's a room like this wherever there's a child, of course,' said the lady. 'How refreshingly ignorant you are! - no, I don't mean ignorant, my dear. Of course, you're awfully well up in ancient History. But I see you haven't done your Duties of Citizenship Course yet.'
'But beggars, and people like that?' persisted Anthea 'and tramps and people who haven't any homes?'
'People who haven't any homes?' repeated the lady. 'I really DON'T understand what you're talking about.'
'It's all different in our country,' said Cyril carefully; and I have read it used to be different in London. Usedn't people to have no homes and beg because they were hungry? And wasn't London very black and dirty once upon a time? And the Thames all muddy and filthy? And narrow streets, and - '
'You must have been reading very old-fashioned books,' said the lady. 'Why, all that was in the dark ages! My husband can tell you more about it than I can. He took Ancient History as one of his special subjects.'
Perhaps women are no longer expected to be 'ladylike', and certainly they're not expected to stay at home and look after the children while their husbands go out to work. But there is still a concept of men and women, who are still distinguishable from each other by the way they look and probably the way they act and dress too.